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NAVAL POSTGRADUATE SCHOOLMonterey, California
DTICAD-A274 903 UAI I 11111IIUiElI IIIIIIIIIIIIlIEttlIIlil • AN2G 1994
THESISA SURVEY OF UNCONTROLLED SATELLITE
REENTRY AND IMPACT PREDICTION
by
Brian D. Neuenfeldt
and
William K. Henderson
September, 1993
Co-Thesis Advisor: I. Michael RossCo-Thesis Advisor: Joseph J. F. Liu
Approved for public release; distribution is unlimited.
94-02283
94 1 2 5 074 4illIlllll!iIIII/IIlI
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4. TITLE AND SUBTITLE: A Survey of Uncontrolled S ý1 -1lite Reentry 5. FUNDING NUMB1ERSand Impact Prediction
5i. AUTIhORS: Neuenfeldt, Brian D. and Henderson, Wilii am K.7. PERFORMING ORGANIZATION NAME(S) AND ADDF ESS(ES) 8. PERFORMING
Naval Postgraduate School ORGANIZATION
Monterey CA 93943-5000)O REPORT NUMBER
9. SPONSORII'IG/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10.Director, Navy Space Systems Division (N63) Naval Space Command SPONSORING/MONITORINGSpace and Electronic Warfare Directorate Code N416T AGENCY REPORT NUMBERChief of Naval Ope:rations Dahigren VAIWashington DC 20350-2000) 22448-51701L11. SUPPLEMENTARY NOTES The views expressed .;n this thesis are those of the author and do notreflect. the official policy or position of tl-e Department of Defense or the U.S. Government.12-A. DISTRI]BUTION/AVAILABILITY STATEMENT 12b. DISTRIBUjIION CODE
Approved for public release; distribution is unlimited.A
STPe C priarygoa o ths tesi i to identify the 'state-of-the-art* in orbit-decay-induced uncontrolled reentry/impact
pre~iction methods, with an emphasis on the physics of the final few revolutions to impact. This was accomplished througha comprehensive literature survey from the 1950's to the present of unclassified military and civil databases. The results ofthe siurvey show that the current U.S. and ititernational reentry/impact prediction methodologies are based on analysis whichis over 30 years old. Of the various 'extensions* to the current reentry theory, of which the NORAD method is recogn~izedas the international standard, there does not appear to be any one method which is singularly superior to the others. It hasalso been shown that numerous reentry investigations made simplifying assumptionis due to innufficierL dzta needed toaccurately model reentry and also because of computing limitations of their day. Also, current deterministic dynamicmodels appear to inadequately describe the actual uncontrolled reentry process, due to a lack of observational data,uncertainty in determining aerodynamic co>efficients, Ltmospberic density, and point mass modeling where changes i"vehicle configuration, attitude and lift are neglected. Stochastic and statistical methods could be applied to the currentniethodology, to better analyz~e the various uncertainties, which could help to improve the over-all predicted impact time aridlocation; however, fusrther research into these methods along with the physics of uncontrolled reeintry is necessay.
14. SUBJECT TERMS: reentry, uncontrolled reentry, reentry effects, reentry prediction, impact 15. NUMBIR OF PAGESprediction, reentry motion, reentry aerotherniodynamics, satellite breakup, atmospheric density 269m,-d2ls,, reentry/impact models 16. PRICE CODE
17. SECURITY CLASStFICA- I8. SECURITY CLASSHFICATION 19. SECURITY CLASSIFICA- 20. LtMITATION OFTION OF REPORT OF Till PAGE TiON OF ABSTRACT ABSTRACTUnclassified Unclassified Unclassified UL.
ýisdad or 19 (cy719rPrwliiý.d Sy ANSI Std. 239-19
Approved for public release; distribution is unlimited.
A SURVEY OF UNCONTROLLED SATELLITE REENTRY AND IMPACTPREDICTION
by
Brian D. NeuenfeldtLieutenant Commander (sel), United States NavyB.S., University of fexas at San Antonio, 1982
and
William K. HendersonLieutenant, United States Navy
B.A., Western State College of Gunnison Colorado, 1984
Submitted in partial fulfillmentof the requirements for the degree of
MASTER OF SCIENCE IN SYSTEMS TECHNOLOGY(SPACE SYSTEMS OPERATIONS)
from the
NAVAL POSTGRADUATE SCHOOLSeptember 1993
Authors: <--- ..Brian D. Neuen eldt
William K. Henderson
Approved by: -__ _---
I. M-icaIl Ross, Co-Th-esis Advisor
-rosepi F Liu, ý,o Th is Advisor
Rudolph Panholze hairmanSpace Systems Academic Gioup
[i
ABSTRACT
The primary goal of this thesis is to identify the "state-of-the-art" in orbit-decay-
induced uncontrolled reentry/impact prediction methods, with an emphasis on the physics
of the final few revolutions to impact. This was accomplished through a comprehensive
literature survey from the 1950's to the present of unclassified military and civil
d~atabases. The results of the survey show that the current U.S. and international
reentry/impact prediction methodologies are based on analysis which is over 30 years
old. Of the various "extensions" to the current reentry theory, of which the NORAD
method is recognized as the international standard, there does not appear to be any one
method which is singularly superior to the others. It has also been snown that numerous
reentry investigations made simplifying assumptions due to insufficient data needed to
accurately model reentry and also because of computing limitations of their day. Also,
current deterministic dynamic models appear to inadequately describe the actual
uncontrolled reentry process, due to a lack of observational data, uncertainty in
determining aerodynamic coefficients, atmospheric density, and point mass modeling
where changes in vehicle configuration, attitude and lift are neglected. Stochastic and
statistical methods could be applied to the current methodology, to better analyze the
various uncertainties, which could help to improve 01- overall predicted impact time and ,,j
location; however, further research into these methods along with the physics of ,, ,
uncontrolled reentry is necessary.DTJ.C QUALITY L-PJ61%CThjj 5 fIy
A) '
II -,,-
TABLE OF CONTENTS
I. INTRODUCTION ..................................... 1
A. HISTORY . .................................... 1
1. Case Studies ...................... ... ...... . 2
a. Sputnik IV .................. . ... ...... . 3
b. Skylab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 4
c. Cosm os 954 ............. ................ 8
B. PROBLEM STATEMENT ............................ 12
C. PURPOSE . ................................... 14
D. METHODOLOG ................................... 15
11. FUNDAMENTALS OF ATMOSPHERIC REENTRY .............. 18
A. TYPES OF REENTRY ............................ 18
1. Uncontrolled Reentry ........................... 18
2. Controlled Reentry ......... ................... 18
a. Ballistic Reentry. ............................. 20
b. Gliding Reentry ........................... 20
c. Skip Reentry .. .............. ........ ... .. 20
iv
B. MAJOR FACTORS INFLUENCING UNCONTROLLED
REENTRY .................................... 21
1. Modeling Reentry Equations Of Motion ................. 22
a. Basic Equations of a Rigid Body .................. 22
b. M odeling Gravity .......................... 23
c. Modeling Atmospheric Drag ..................... 24
(1) Modeling Atmospheric Density ............... 28
(2) Modeling Aerodynamic Coefficients .. ......... 33
(3) Rarified Gas Dynamics ................... 33
(4) Vehicle Profile Area ...................... 36
(5) Vehicle M ass .. ....................... 37
(b) Vehicle Velocity. ....................... 37
d. Modeling Aerodynamic Lift ................... 37
e. Reentry Equations of Motion ................... 38
(1) tundamental Equations of Entry Dynamics ...... 39
2. M odeling Brcakup ............................. 43
a. Reentry Body Structural Mechanics ............... 43
b. Modeling Reentry Heating Effects ..... ............ 44
(1) Reentry Heat Input ....................... 46
(2) Reentry Heating Rate ...................... 51
(3) Ablation .. ........... .............. 51
C. GENERAL DESCRIPTION OF THE REENTRY PHASE ........ 52
V
D. CURRENT REENTRY THEORIES ....................... 55
1. Physical Modeling Theory ........................ 57
2. Mean Motion Theory ........................... 62
IIl. FORMULATIONS AND SOLUTIONS OF REENTRY .............. 66
A. ANALYTICAL REENTRY EQUATIONS OF MOTION ......... 66
1. Chapman's Approximate Analytical Entry Equations of Motion . 66
2. Loh's Second Order Unified Solution of Entry Dynamics ..... 71
3. Yaroshevskii's Entry Theory ........................ 74
4. Universal Equations for Orbit Decay and Reentry .......... 78
5. Attitude Dynamics *of Uncontrolled Motion During Reentry . . . . 86
a. Equations of Perturbed Motion ................... 93
b. Effect of Motion Relative to the Center of Mass on the
Motion of the Center of Mass ................... 103
c. Follow-on Investigations of Uncontrolled Reentry Body
M otion . . ............... . .. .. ... . ... ... 106
B. SIX-DEGREE-OF-FREEDOM SIMULATIONS .............. 107
C. RAREFIED GAS DYNAMICS ......................... 111
1. Newtonian Aerodynamics for Hypersonic Continuum Flow . . .. 117
2. Free Molecular Flow Model ......................... 123
3. Bridging Free Moleular and Continuum Flow ............. 126
a. Shape Element Bridging Method .................. 126
vi
b. Local Bridging Method ...................... 131
4. Gas-Surface/Gas-Gas Interactions ...................... 132
D. SURFACE ROUGHNESS EFFECTS ...................... 136
E. REENTRY HEATING EQUATIONS .................... 144
F. STRUCTURAL BREAK:- OF A REENTRY BODY ......... 146
IV. DETERMINISTIC REENTRY/IMPACT PREDICTION METHODS .... 153
A. CURRENT REENTRY/IMPACT PREDICTION METHOD ...... 153
B. ALTERNATE REENTRY/IMPACT PREDICTION METHODS ... 156
1. Reentry Prediction Methods At ESOC ................. 156
2. The LIFETIME Model ........................... .162
3. Mod1eing Ballistic Cefficient ... . . . .. 171
C. FACTORS INFLUENCING REENTRY/IMPACT DISPERSION ... 182
V. STOCHASTIC AND STATISTICAL PREDICTION METHODS ....... 190
A. EXTENSIONS OF THE PHYSICAL MODELING REENTRY
TH EORY ..................................... 190
1. Estimation of Reentry Trajctonies .................... 190
2. Analysis of Tracking and Impact Prediction (TIP) .......... 202
B. MONTE CARLO ANALYSIS OF SKYLAB'S IMPACT AREA .. . 214
1. Simulation Results ............................. 221
vii
VI. CONCLUSIONS AND RECOMMENDATIONS ................. 233
A. CONCLUSIONS ... ............................ 233
B. RECOMMENDATIONS ............................ 238
LIST OF REFERENCES . ................................. 240
INITIAL DISTRIBUTION LIST .............................. 249
viii
LIST OF FIGURES
Figure 1: Sputnik IV Final Revo!ution[Ref. 9] ............................................... 5
Figure 2: Sputnik IV Impact Area/Debris Location[Ref. 9] . . . . . . . . . . . . . . . .. .. . .. . . . . . . . .. . .. . . . . . . . . . . . 6
Figure 3: Sputnik IV Impact Area Ground Trace,[Ref. 9] . . . . . .. . . . . . .. .. . . . . .. . . . . . . . .. . .. . . . . . . . . . . . 7
Figure 4: Cosmos 954 Impact Area / Debris Dispersion[Ref. 10] . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . 9
Figure 5: Skylab Final Ont.e-Quarter Revolution[Ref. 11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 6: Types Of Reentry[Ref. 14] . . . . . . . . . . . .. ... . . . . . . . . . .. . .. . .. . . . . . . . . . . . 19
Figure 7: Eanth Geopotentia! Model[Ref. 17] . ... ... ... ... ... ... ... ... ... ... .. ... ... . . . .. .25
Figure 8: Drag Effect On High Eccentricity Orbits[Ref. 17] . . .. .. .... ... . .. ... ... ... ... ... .. ... ... . .. .. 27
Figure 9: Flow Regimes[Ref. 20] . . . . ... .. . ... ... .. . .. . . .. .. . .. . .. . .. .. . . . . .. 35
Figure 10: Inertial Coordinate System[Ref. 40] ................................................. 40
Figure 11: Relationship Between Relative And Inertial Velocity[Ref. 40] ................................................. 41
Figure 12: Changes During Atmospheric Reentry[Ref. 43] .. ........................................ . . 54
Figure 13: Maximum Deceleration vs Initial Flight Path Angle[Ref. 14] . . . . ... ... . .. ... .. . .. . . . . .. . .. . .. . . . . . . . . . .. 56
ix
Figure 14: Differential Corrections Display[Ref. 49] ................................................... 60
Figure 15: Tracking And Impact Prediction (TIP) Display[Ref. 49] .................................................. 61
Figure 16: Z Function Solution Graph For L/D1=0[Ref. 14] ................................................... 69
Figure 17: Z Function Solution Graph For Various L/D's[Ref. 14] .. . .. . .. ... .. . .. . .. . .. . .. ... . .. . .. . .. . . . .. . . 70
Figure 18: Loh's Second-Order Range Of Applicability[Ref. 40] .. . .. ... ... .. . .. . .. . .. . .. ... . .. . .. . . . . .. .. . . 72
Figure 19: Loh's Second-Order vs Chapman's First-Order[Ref. 40] .. . .. . .. ... .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . . . 75
Figure 20: Coordinate System And Nomenclature[Ref. 41] . . . . . . . . .. . .. . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 79
Figure 21: Aerodynam:c Force Diagram[Ref. 41] .................................................. 80
Figure 22: Variations Of --y vs The Nondimensional Velocity (v)[Ref. 41] .................................................. 87
Figure 23: Variations Of ln(Z/74_ vs The Nondimensional Velocity (v)[Ref. 41] . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . 88
Figure 24: Variations Of G vs The Nondimensional Velocity (v)[R ef. 4 1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Figure 25: Attitude Dynamics Coordinate System[Ref. S7] .. ... .. . .. . .. . .. . .. . .. ... ... . .. . .. . . . . . . .. . . 92
Figure 26: Trajectory Of The Reentry Body[Ref. 56] .. ... ... .. . .. . .. ... ... ... ... ... ... ... .. . .. . . 95
Figure 27: Reentry Body Coordinate System[Ref. 56] .. ... ... ... .. . .. ... ... ... ... ... ... ... .. . .. . . 97
x
Figure 28: Six-Degree-Of Freedom Simulation Flowchart[Ref. 62] .. . . .. . .. . .. . .. .. . . . ... . .. .. . .. . . . . . . . . . . .. . 108
Figure 29: Simulated Generic Spacecraft And Reactor Sub-Element[Ref. 64] ................................................... 110
Figure 30: Flow Regimes - Altitude vs Velocity[Ref. 76] ................................................... 112
Figure 31: Aerodynamic Coefficients In The Flow Regimes[R ef. 76] . .. .. . . .. .. . ... .. . . . .. . . . . ... .. . .. . . . . . . . .. . 114
Figure 32: Shape Elements And Composed Bodies[R ef. 76] . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 116
Figure 33: Shape Factor As A Function Of (d/rN)[R ef. 76] . .. . . . . .. . . . . .. .. . . . . . . . . . .. . . . . . . . . . . . . . . . . 119
Figure 34: Newtonian Lift And Drag Functions[R ef. 76] . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 120
Figure 35: Sphere Drag Coefficient In Rarefied Flow[Ref. 76]...........................127.=[R f 76 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Figure 36: Disk Drag Coefficient In Rarefied Flow[Ref. 76] ................................................... 128
Figure 37: Molecular Velocity Distribution[R ef. 80] . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Figure 38: Average Translational Energy[Ref. 80] . .. ... ... ......... ........ .. ... ... . .. . 137
Figure 39: Boundary Layer Types[Ref. 22] ...................................... ..... 140
Figure 40: Boundary-Layer Velocity-Distance Profiles[R ef. 22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 41: Altitude-Air Speed-Dynamic Pressure Relationship[Ref. 22] . .. . .. . .. . .. ... .. . . . . .. . . . . .. .. . . . . . . .. . . .. . 142
xi
Figure 42: Structural Breakup - Heating Rate vs Altitude"IRef. 861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Figure 43: Breakup Altitudes For Conected Heating Rates(Ref. 64) ................................................... 150
Figure 44: Salyut-7/Cosmos-1686 Final Descent Altitude Profile[Ref. 94] .................................................. 163
Figure 45: LIFETIME Debris Dispersion Footprint[Ref. 95] ................................................... 168
Figure 46: LIFETIME Groundtrack And Altitude Decay History[Ref. 95] ................................................... 169
Figure 47: Best Fit DC RMS For Cosmos-954/1402[Ref. 92]. .................................................. 177
Figure 48: Dependence Of "1 ve, Latitude And Longitude On B[Ref. 471 ............................................ 181
Figure 49: Near-itarth View Of All Cataloged Space Objects (1987)[Ref. IGO] .................................................. 186
Figure 50: U.S. Radar And Electro-Optical Assets[Ref. 100] .................................................. 187
Figure 51: Orbital Plane Rotation Due To A RotatiaLg Atmosphere[Ref 101] .................................................. 189
Figure 52: Tangent Plane Projection Of Earth Impact Location[Ref. 1021 .................................................. 200
Figure 53: Decay Predicted Accuracy (By Year)[Ref. 48] .................................................. 203
Figure 54: Final Run vs Vis Obs Mean 'rime Error[Ref. 48] .................................................. 205
Figure 55: Final Time Error Standard Deviation[Ref. 48] .................................................. 206
xii
Figure 56: Mean Time Error And Stndard Deviation (1987-1990)[Ref. 481 .......... .............. ........ ......... ... 208
Figure 57: Mean Location Error And Standard Deviation (1987-1990)(Ref. 48] ........................................... 210
Figure 58: Regression Model Mean Approximate Error (1987-1990)[Ref. 48] ... ........................................... 213
Figure 59: Numerical Stability Region[Ref. 45] ..... ..... .............. ........ ... ... ... ... 219
Figure 60: Relative Entry Flight Path Angle Scattergramr[Ref. 45]. .. ... ... .. ...................... ...... ... ... 223
Figure 61: Skylab Nominal Flight Characteristics[Ref. 45] ............................................ 224
Figure 62: Downrange Impact Point Variations[Ref. 45] .. ... ... .. .. ....... ... ... ... .. ... ... ... ... .. . 225
Figure 63: Latitude Histcgramn / Cumulative Probability Distribution[Ref. 45] ............................................ 227
Figure 64; Longitude Histogram / Cumulative Probability Distribution[Ref. 45] . ...... ... .. ................. ..... ... ... ...... 228
Figure 65: Down Range And Crossrange Dispersions[Ref. 45] . . ... .. . .. .. . .. . .. . .. . . . .. . .. . . . . . . . . . .. . .. . 231
Figure 66: Downrange And Uprange Dispersions Three-Sigma Limits[Ref. 45] .. ... .. . .. .. ... . . . . . . . .. .. . .. . . . .. . . . . .. . .. . 232
xiii
LIST OF TABLES
Table I: SKYLAB RECOVERED DEBRIS (PARTIAL LISTING)"Ref. 101 . . .. . .. . . . . ... ... .. . .. . .. . . . . .. ... .. . .. . .. . . 10
Table 11: OBSERVATIONAL DATA FORMAT[Ref. 49] . . ... .. . .. .... ...... .. ... ... .... .. .. . .. . .. .. 59
Table III: BRIDGING DEPENDENCIES[Ref. 76) . . ... ... .. ....... .. ... ... ... ... ... .. . .. . .. .. 115
Table IV: TRANSITIONAL BRIDGING FUNCTIONS[Ref. 76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Table V: NEWTONIAN LOCAL PRESSURE LAW[Ref. 76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Table VI: SHAPE FACTOR VALUES[Ref. 76] . . .. . . . . . . . .. . . .. .. . .. . .. .. . .. . . . . .. . .. . .. .. 121
Table VII: BODY SHAPE AND LIFT SLOPE[Ref. 76] . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Table VIII: LOCAL SURFACE PRESSURE AND SHEAR STRESS[Ref. 76] . . .. . . . .. . ... . .. ... . .. . . . .. . . . . .. . . . . .. . . . .. 124
Table IX: AERODYNAMIC FORCE COEFFICIENTS[Ref. 76] . . .. .. . .. . . .. ... ... . .. ... .. . .. . . . . . . . .. . . . .. 124
Table X: SHAPE DEPENDENT BOUNDARIES[R ef. 76] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Table XI: FREESTREAM CONDITIONS[Ref. 80] . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. 136
Table XII: RESULTS OF BREAKUP ANALYSIS[Ref. 6 ]A] . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Table XlII: SALYUT-7/COSMOS-1686 FINAL DECAY PREDICTIONS[Ref. 94] . ..... .... ... ... ...... .. . .. ...... ... ... .. . .. 164
xiv
Table XIV: DISTRIBUTION OF DECAYED ORBITAL OBJECTS[Ref. 92] .. .... ............ .......................... 173
Table XV: DECAY TIME ERROR RMS[R ef. 92] . .. ... ..... .... .. ... .. ... .. ... ... .... .. ... .. 178
Table XVI: ENTRY STATE VECTOR / ERROR COVARIANCE MATR1X[R ef. 45] . .. .................................... ..... 222
xv
ACKNOWLEDGEMIENTS
I would like to thank some of the many people who helped make my time at the
Naval Postgraduate School one of the most rewarding tours of my naval career as well
as personally fulfilling. First, there could be no better co-thesis advisors than Dr. 1. M.
Ross and Dr. J. J. F. Liu. Wihout the tireless efforts of these two scholars and subject
matter experts, this thesis would have been n~early impossible to complete. Secondly, 1
would like to thank the Space Systems Operations academic advisor, Dr. D. C. Bo0ger.
I will never forget his advice to me at the beginning of my pcstgra-duat, A.ucLI.jon,
".. .take all the math you can, it can't hurt you," he was right! Next, I would like to
acknowledge the superb academic staff of the Naval Postgraduate School without whom
an unde~rprmdtiate biology major could never have been transformed into a soace systems
operations major. And of course, this thesis is only half my work.. Without the daily
exchange of ideas and suggestions of my thesis partner and good friend, LT Kelly
Henderson, this would never have happened. Finally, I must thank my family. I could
never have made it this far in my career without the constant support of my parents and
parents-in-law, especially when I was at sea and my wife was alone with the boys. And
so it goes without saying that my wife, Mary, is my driving force and my children
(Matthew, Christopher, Timothy and Mark) are my inspiration. I could not have
completed this course of study or the thesis without their endless support and love.
Brian D. Neuenfeldt
Xvi
ACKNOWLEDGEMENTS
I would like to sincerely thank all of those individuals who have helpeA i my
efforts. Since the beginning of my academic career at the Naval Postgraduate School,
many prominent people have given their valuable time and expertise which has made this
tour, both professionally and personally, one of the most meaningful and rewarding tours
in my Naval career. I owe a great deal of gratitude to my co-thesis advisors, Dr. I.
Michael Ross and Dr. Joseph J. F. Liu, for their expert guidance and wisdom. Their
willingness to invest a significant amount of time and effort into making this a
meaningful thesis is truly appreciated. Also, I would again like to thank Dr. John
Darrah and Dr. Liu's staff at the Air Force Space Command, Dr. Steve Knowles and
his staff at the Naval Space. CommLno, Mr Frank Marcon t ?I: t Air Pnrrp ,lpnnlhuirc
Laboratory and Dr. George Chao at The Aerospace Corporation for their technical
support. To my partner and good friend, LCDR Brian Neuenfeldt, it has been a pleasure
working with you. Your suggestions, ideas and unique approach have greatly contributed
to strengths of this thesis. I would also like !o thank my parents and parents-in-laws for
supporting my family and I throughout my career. Finally, my greatest thanks goes to
my wife and life long partner, Monica, and my children, Zeb and Jeremiah. Without
their unselfish love and support, this academic and final endeavor would not have been
possible.William K. Henderson
xvii
1. INTRODUCTION
A. HISTORY
Satellite or spacecraft reentry related work has been done in this country for nearly
five decades. With the advent of the rocket in World War II, the military saw the first
application of spacecraft reentry prediction as the ballistic trajectory calculation of the
target aimpoint. Prior to this, several nations had "rocket societies," however, none had
actually applied the use of "space" to a technology or an industry. Thus, the first
application of space was by the military for the purpose of war. [Ref. l:p. 13]
On October 4, 1957, the former Soviet Union changed the perception of space held
by most Americans and possibly the world. It was the launch of Sputnik I that caused,
then Senator, Lyndon B. Johnson to remark:
That sky had always been so friendly, and had brought us beautiful stars andmoonlight and comfort; all at once it seemed to have some question marks all overit. [Ref. 2:p. 13]
When the former Soviets again launched another rocket barely on-- month later and
sent a dog into space, onboard Sputnik II, the world now saw, the first space traveler and
the dreara of humans in space became more real than dream [Ref. l:p. 20].
Afte: a dismal failure of the United States' first attempt to launch a satellite into
orbit with Vanguard I, which was dubbed by some reporters as "Kaputnik" after it
exploded on the launch pad on December 16, 1957, Explorer I was successfully launched
1|
on January 31, 1958. The space race had begun, and both countries were pushinz to
have the first human in space. [Ref. l:p. 20], [Ref. 2:p. 16]
With the goal of putting humans into space came the necessary requirement of
providing for the safe return of those humans back to Earth, unlike the first -log in space
which perished after about a week when the oxygen supply was exhausted [Ref. 2:p. 16].
This was the beginning uf the most intense research period into the reentry process over
the entire history of space exploration.
By April 1972, there were at least 44 reported instances where man-made space
objects had impacted on the Earth [Ref. 3:D. 383]. By March 1978, the total count of
man-made objects placed into Earth orbit was 10,791 [Ref. 4:p. 107]. By Aug 1991, the
cumulative count of objects ever placed into space was 21,231 with 14,417
decays/reentries, leaving 6,814 objects in Earth orbit [Ref. 5:p. v).
The return or reentry of manned spaceflights has been covered by the various
media sources with varying degrees of intensity base-d upon the "newsworthiness" of the
event and sensed public interest:
Solar Max satellite plungcs to Earth. [Ref. 6:p. 21(N)]
Spacecraft's study of sun ends tod y: Solar Max heads for fiery reentry. [Ref.7:p. A3]
If you see a shooting star Dec. 8, make a fast wish for a deep cave. [Ref. 8:p. B1]
1. Case Studies
It is useful for the reader to understand the motivation for studying the
reentry process. Obviously, the routine recovery of numerous U.S. and Soviet spacecraft
2
must imply that the reentry process is adequately understood. In the case of ccntrolled
reentry and space. vehicles designed for reentry, this may well be the case; however, in
the uncontrolled reentry case, this is not as easy to conclude.
a. Sputnik IV
September 5, 1962, the first recorded impact of a man-made space object
in the United States was satellite number, 1960 el, Sputnik IV. This satellite reentered
the Earth's atmosphere over North America and many fragments of the reentering debri-
ended their orbit over the state of Wisconsin. The largest fragment of which was found
to weigh approximately 21 pounds (9.49 kg) impacted in the city of Manitowoc,
Wisconsin at approximately 0530 local time. [Ref. 9 :p. 1]
Sputnik IV, which had been launched on May 14, 1960, was designed
to test life-support systems for Soviet manned space flight. On May 19, a planned
deorbit maneuver failed and the pressure vessel separated from the cabin, leaving the
cabin in a "lopsided" orbit. [Ref. 9:p. 2] There were a total of nine separate orbiting
objects associated with satellite 1960 el. The first of these objects to reenter was the last
stage rocket body, which reentered on July 17, 1960. By July 1, 1961, six of the nine
pieces had reentered. The payload reentered on September 5, 1.962 leaving only one
other object in orbit. [Ref. 9:p. 3]
The U.S. Space Detection and Tracking System (SPADATS) predicted
Sputnik IV would reenter on or about September 6, 1962. Moonwatch observers, teams
of volunteers who attempted to observe reentry events with either the naked eye or
telescopes, were notified of the SPADATS prediction on August 29, 1962. [Ref. 9 :p. 10]
3
It was based on these observations that the fragments of Sputnik IV were eventually
located, with the exception of the largest fragment, which was found by a routine police
patrol of the city of Manitowoc. Figure 1 shows the satellite ground trace over the last
one-half revolution [Ref. 9:p. 20]. Figure 2 shows the ground trace of the last one
hundred nautical miles [Ref. 9 :p. 121. Figure 3 shows the impact location of objects
recovered in Markitowoc [Ref. 9:p. 17]. The objects found on the roof of the church
annex, noted in Figure 3, consisted of 15 small spherules approximately 1/8 inch in
diameter. These were about 325 feet further downrange from the initial mass which
impacted near the intersection of Park St. and 8th St.
There was no loss of life and the only reported property damage was the
impact impression left in the street.
b. Skylab
In May 1973, the last U.S. Saturn V rocket launched Skylab into orbit
approximately 270 nautical miles (nm) above the Earth. Skylab was the first U.S. space
station and the centerpiece of the U.S. space program since the last U.S. moon mission.
Skylab would remain in orbit until July 11, 1979. [Ref. 10:p. 1]
Knowing the inevitability of decay and reentry of low Earth orbits,
NASA contracted the Lockheed Missile and Space Company (LMSC) to investigate the
predicted reentry and breakup of Skylab three years prior to its launch. This initial
report predicted that Skylab would begin to breakup at an altitude of 65 nm (120 kin) and
that debris would fall 3600 nm downrange from the initial breakup. [Ref. 10:p. 2]
4
I. ".-- -'I? - : i. ,,,,. ,p . ,
S- " I '" I 4
-51
II -- - -- - - - - I rlp
4)• ., I, 1
[Ref 9fI[
T I
Y I I -s_-
[Rf -9].i'
Figre2:SptnkIIpatAelersLcio[Ref.49
N A 46
AA
I L ITl
lieof Rev. Andersen~
HouseU
I'irst Luthe.ran Churc.1
Annex aK
all
Figure 3: Sputnik IV Impact Area Ground Trace[Ref. 91
7
Based on the post-flight reconstruction of the reentry of Skylab, Marshall
Space Flight Center (MFSC) was able to conclude that telemeti y iignals were still being
sent from Skylab as it passed over Bermuda and Ascension Islands. Also, each of these
tracking stations reported only one radar contact. It was therefore concluded that Skylab
was intact and breakup had not begun although the altitude had decreased to 57 nm. [Ref.
10:p. 5] Survivability underestimation has been typical of past analysis as will be
discussed in Chapter III.
Figure 4 shows the ground trace of Skylab over the last one-quarter
revolution and denotes the irrmpact footprint (debris dispersion area) [Ref. 10:p. 18].
T~ble 1 is a partial listing of recovered debris from Skylab and the recovery location
[Ref. 10:p. 10].
Again, there was no loss of life, no human injury and no property
damage as a result of' this uncontrolled reentry.
c. Cosmos 954
January 24, 1978 marked the first uncontrolled reentry and Earth impact
of a nuclear powered artificial satellite [Ref. 3:p. 384]. This was also the first case of
"significant" property damage due to an artificial Earth satellite impact. It was reported
that Canada spent over 11 million dollars and the Unitcd States spent nearly 3 million
dollars in the location and recovery of radioactive debris [Ref. 3:p. 386]. Figure 5
shows the reentry ground trace and impact dispeision area of Cosmos 954 [Ref. 11 :p.
303].
8
ww
I-z
I-7
Figure 4: Cosmos 954 Impact Area /Debris Dispersion[Ref. 10]
9
Table 1: SKYLAB RECOVERED DEBRIS (PARTIAL LISTING)[Ref. 10]
Charred Fragoents OwS 33.9S, 121.9E (In Experence)
Burned Material OWS 33.9S, 121.9E (In Esperance)
Aluminum 356 Canting OWS 33.76, 122.1E (20 ml NE ofEsperance)
Foam riberglass OWS 33.9S, 122.0E (9 mi E ofBeam Section Esperance)
H20 Tank ows 33.8S, 122.0E (9 mi Nv. of
,.ft End Esperance)
i1f0 Tank OWS 33.9S, 122.1E (10 r'i E. ofEi.pera~n'ce)
10' Steel Strip OWS 33.9S, 122.3E (25 mi E of(H20 Tank) Esperance)
Heat Exchanger OWS 33.9S, 122.1Z (12 ml E. of(H2 0 Cooler) Esperance)
Segoint of OWS 33.9S, 122.1E (11 ml E oiPiberglass Sphere Esperance)
Insulation OWS 33.9S, 122.1E (11 wi E of(Bulkhead) rsperance)
Aluminum Gear OHS 33.7S, 122.5E (40 zi WE ofand Housing (Urine Separator) Espezance)
N Tank AM 33.2S, 122.6E (60 ul NE ofA EsperancG)
Electronics Module AM 33.5S, 122.3E (35 ml NE ofEmperance)
N2 Sphere AM :!3.5S, 122.8Z (49 Mi ENE of2eperanoee inNeridup %rea)
Pressure Tank IU 33.2S, 122.6E (60 ml NE ofEsperan ze)
]C)
r •f
U I
cc .1a ._ u,2
_
-JJ
-3-
2 -
Figure 5: Skylab Final One-Quarter Revolution[Ref. 111
ll1
The Outer Space Treaty of 1967, the Liability Convention of 1972 and
the Registration Convention of 1976 are the three principal documents, drafted by the
United Nations Committee on the Peaceful Uses of Outer Space and its Legal
Subcommittee, which define ownership responsibilities of space objects [Ref. 12 :p. 457].
Although the Outer Space Treaty, Article VIII, provides in explicit terms the
requirements that the registry state (country of origin) retains jurisdiction and control
over its satellites while in space or on a celestial body, [Ref. 3:p. 389] there may be no
legal requirement under the Liability Convention or the Rescue and Return Agreement
for the former Soviet Union to reimburse either the U.S. or Canada for the cleanup cost
of Cosmos 954 [Ref. 3:p. 387].
It is because of the potential for the loss of human life, significant
property damage and legal responsibility based in international law that the reentry of
uncontrolled artificial satellites must be further investigated. In the case of Cosmos 954,
on the morning of reentry, the Soviets predicted reentry impact near the Aleutian Islands
(520 N,173°W) and the North American Air Defense Command (NORAD) predicted
reentry impact near Hawaii (19'N, 156°W) [Ref. 4:p. 110]. The physics of uncoatrolled
reentry and the modeling of the reentry process must be better understood in order to
improve the accuracy of reentry predictions.
B. PROBLEM STATEMENT
The probability of space objects reentering the Earth's atmosphere and surviving
to impact is -elatively small; most artificial objects burn up in the atmosphere before they
12
impact [Ref. 13:p. 1]. Falling debris nonetheless presents a potential hazard to people
and property. The most difficult problem associated with reentry impact prediction is
not to determine what will survive to impact in as much as where impact will occu,.
[Ref. 5:p. v]
The significant factors related to the prediction of time to impact and location of
the reentry are a lack of observational data over the entire orbit and a lack of "precise"
mathematical models which accurately describe the physical processes occurring during
reentry. The major parameters which contribut. to the uncertainties of the reentry
prediction are: [Ref. 5:p. v]
1. Atmospheric density variations--atmospheric density is strongly influenced bysolar and geomagnetic activity, both of which are difficult to forecast.
2. Aerodynamic force models--aerodynamic forces are a function of attitude, liftand drag coefficients, gas-surface interactions and gas dynamics such ascontinuum flovw or free-flow regimes.
3. Spacecraft attitude motion--the attitude of the object and how it changes withtime is an important factor in estimating the aerodynamic forces encounteredduring reentry.
4. Changes in configuration--ablation and fragmentation cause changes incontfiguration (profile area and mass/area loss) which can change theaerodynamic forces experienced either as an increase or decrease in net force.
The lack of regularly spaced observational data over the entire orbit can severely
handicap the effoits to predict reentry, especially in the final phase. Since most
observational data is from radar tracking stations and the object is in a very low orbit,
the time in view is of short duration as well as limited by t) e geographic location of the
13
tracking stations. The availability of good quality tracking data is required for accurate
reentry predictions and could compensate for inherent deficiencies in the models. [Ref.
5 :p. vi]
C. PURPOSE
This research was initiated by the Air Force Space Cor nand. The purpose of this
thesis is threefold:
1. Conduct a comprehensive literature survey in the area of artificial satellitereentry specifically, uncontrolled orbit decay and reentry into the Earth'satmosphere.
2. Describe the "state-of-the-art" of reentry/impact prediction techniques.
3. Define critical areas of research where increased emphasis is required in orderto improve the accuracy of the reentry prediction.
The focus of this thesis is the physical processes of the uncontrolled reentry from
the final few revolutions to impact. It is not the intended purpose of this thesis to
examine in detail or focus on the following aspects of the reentry prediction problem:
1. Atmospheric density models
a. Solar activity and influence on reentry
b. Geomagnetic influence c-n reentry
2. Long-.term orbital lifetime prediction
14
It is necessary, however, to discuss these aspects of th: reentry process in order to
completely study the physical phenomena associated with reentry. These topics are the
subject matter of Chapter II and are considered the fundamental background material
required for a further, more detailed study of the reentry process.
D. METHODOLOGY
The primary research methodology of this thesis was a comprehensive literature
survey of military and civil aerospace data bases. The literature search of Department
of Defense (DoD) data bases was restricted to unclassified work. The literature search
was conducted at or through the following activities:'
1. Naval Postgraduate School, Dudley Knox Library
2. 1.-nsccom Air Fnrce RBse PResercrh hibrn rv
3. University of Colorado at Boulder, Astrophysics and Engineering libraries
4. AFSPACECOM Astrodynamics Division (CNY) technical library
5. United States Air Force Academy Library
The secondary research methodology of this thesis was p'ýrsonal interviews wivh
"experts" in the study of reentry or reentry related fields. The activities which were
contacted or visited personally include:
1. TRW Corporation
2. The Aerospace Corporation
15
MOM-
3. Phillips Laboratories
4. Air Force Geophysics Laboratory (AFGL)
5. Naval Space Surveillance Center (NAVSPASUR)
6. Air Force Space Command (AFSPACECOM)
7. United States Space Command (USSPACECOM)
The primary data bases and periods over which the literature survey was conducted
include:
1. NASA 1940- 1993
2. DTIC 1950- 1993
3. IAA 1960- 1993
4. ST'AR 1960- 1993
5. DIALOG 1970 - 1993
These data bases were searched with similar strategies in an effort to pull as many
"original" articles as possible, while concurrently verifying the search process by
producing numerous "duplicate" articles found in other data bases. The primary search
terms included: reentry, reentry prediction, satellite reentry, atmosphere reentry,
spacecraft, spacecraft reentry, v'ncontrolled reentry, reentry impact prediction, satwlite,
16
descent trajectory, atmosphere drag, atmospheric density models, satellite orbit decay,
reentry heating, reentry dynamics and reentry effects.
In order to prevent confusion of common variables when derived in multiple
sources, the equations and variable nomenclature presented throughout this thesis are as
presented in the original works, with the exception of minor changes made as noted. An
example of such a change is the flight path angle, -y, presented in Chapter II. Since this
is a survey of the literature, which dates back to the 1950's, the authors have made a
conscious effort to preserve the flavor of the individua! works and no attempt has been
made to standardize the nomenclature. However, this is an object for further study as
indicated in Chapter VI.
17
H. FUNDAMENTALS OF ATMOSPHERIC REENTRY
A. TYPES OF REENTRY
The reentry trajectories of space vehicles can be classified into two types, either
uncontrolled or controlled. The focus of this thesis is to investigate the uncontrolled
reentry of satellites, however, three major types of controlled reentry, ballistic, gliding
and skip, as shown in Figure 6, will be described for completeness [Ref. 14:p. 6).
1. Uncontrolled Reentry
Uncontrolled reentry may be the result of an unrecoverable satellite subsystem
failure or the end of the satellite's operational life. The flight path angle is usually much
less than one degree and lift is considered negligible. Typically, very large uncertainties
in impact point predictions are created by decay-induced uncontrolled reentries. [Ref.
15:p. 44]
2. Controlled Reentry
During a controlled reentry, the vehicle's aerodynamic and heating loads are
maintained within acceptable limits by controlling the effects of lift and drag forces on
the vehicle throughout the flight. This is accomplished through a carefully designed
space vehicle, flight trajectory and possibly a precision guidance system. Controlled
reentry spans an aerodynamic flight regime from subsonic to Mach 25 and beyond for
hyperbolic reentry. [Ref. 16 :p. 231]
18
-. .-- Trajectory
Portion of trajectory overwhich analysis is cppJicob!e
BallslicDecoying orbit
[R f "141/ "
9"-
.'" 'i"'. *,.7 :
Glide Skip
Figure 6: Types Of Reenutry[Ref. 14]
19
a. Ballistic Reentry
A ballistic reentry is characterized by sufficiently steep reentry angles
where the force of lift is assumed to be negligible [Ref. 14:p. 2]. The ability to control
the reentry velocity, flight path angle, ballistic coefficient and atmospheric properties
determines the accuracy of ballistic reentry vehicles. Impact accuracies within the
intended target vary from 20 km for shallow angle reentries, such as Mercury type
vehicles, to an accuracy of a few hundred meters for a steep angle reentry of an
intercontinental ballistic missile type vehicle. [Ref. 16:pp. 237-242)
b. Gliding Reentry
A gliding reentry is characterized by a glide slope rather than a reentry
trajectory. During a gliding reentry, a vehicle such as the space shuttle creates enough
lift to maintain a hlong hlyprsoi-ic ide at a srmalt 'All flig•ht• +6 ... p ngle [L .Re.f.. 16 :. 121- A
measure of the vehicle's lift that influences the descent path and cross range capability
of a vehicle is the lift-to-drag (L/D) ratio [Ref. 15:p. 47]. By adjusting the vehicle L/D
ratio or bank angle, a, (the angle between the vehicle lift vector and the plane containing
the vehicle position and velocity vectors), the extent of the range and cross-range can be
controlled [Ref. 15:p. 44].
c. Skip Reentry
A skip reentry is characterized by a vehicle whose L/D is greater than
zero. Sufficient lift is produced to dominate the gravitational and centrifugal forces. If
this lift is combined with a large enough initial angle of descent, a reentry trajectory with
20
one or more skips may be produced. As the vehicle begins to reenter, it reaches a
minimum altitude where it begins to "pull-up" due to the lift force dominating over
gravity. Eventually, the vehicle will exit the atmosphere at a reduced velocity. If the
exit ielocity and flight path angle are correctly controlled, the vehicle will achieve a brief
orbital phase followed by a second reentry dc;,.range from the first. [Ref. 16:pp. 245-
2461
B. MAJOR FACTORS INFLUENCING UNCONTROLLED REENTRY
It is the focus of this thesis to dt .-ribe the physical processes of the final few
revolutions of a reentry body's orbit and the reentry phase to impact. Reentry trajectory
techniques differ from orbit determination techniques. The reentry phase is very
dynamic as opposed to the exoatmospheric phase of orbital motion. Specifically, the
reentry phase is characterized by:
1. Rapidly decreasing altitude
2. Rapidly increasing aerodynamic heating effects
3. Rapidly changing aerodynamic load effects
The reentry phase can be characterized by that portion of the trajectory prior to
brcakup and after breakup. Prior to breakup, the reentry trajectory equations of motion
include parameters such as gravity, atmospheric density, ballistic coefficient, position and
velocity. Solutions to the reentry equations of motion can be found using analytical,
21
semi-analytical and numerical techniques. Breakup can be described as that point in the
trajectory where heating effects and load effects cause the reentry body to lose its
structural integrity.
1. Modeling Reentry Equations Of Motion
a. Basic Equations of a Rigid Body
The three basic equations of a rigid body are
dr V (1)dt
dK F (2)dt
dH =M (3)dt
whereý
H = reentry body's ang-ilar momentum vector relative to its center of mass
K = reentry body's linear momentum vector
M = total moment force vector relative to the center of mass
F = total force vector acting on the body
r = position vector of the body
V = velocity vector of the center of mass
Equations (1) and (2) are the kinematic and force equations, respectively.
When these equations are coupled, they yield one second-order, non-linear, vector
22
differential equation which describes ihree-dimensional motion of the reentry body's
center of mass, given by
d 2r 1 (4)
where g is the gravitational field and a, represents the atmospheric drag acceleration.
It is atmospheric drag that causes an artificial satellite to decay and reenter.
Equation (3) describes the motion of ? body about its center of mass,
commonly referred to as attitude motion. The reentry body's attitude is related to angle
of attack, a, and is an important parameter which will be discussed in the next section.
When the system of equations (1) through (3) are coupled along with the
attitude parameters, it yields two second-order, non-linear, vector differential equations
or six second-order scalar differential equations. These six differential equations
completely describe the six-dimensional motion of arid about the center of mass. This
system of equations is commonly referred to as the six-degree-of-freedom model which
will be described further in Chapter Ill.
b. Modeling Gravity
In equation (4) the gravitational field, g, may be written as an inverse
square relationship
,q -_ (5)r3
23
where
14 = gravitational constant (Earth = 3.9865 x 10' km3/sec 2)
or more accurately via the geopotential model. The geopotential model divides the Earth
into three sets of geographically divided regions described by latitude and longitude as
shown in Figure 7 [Ref. 17:p. 233]. The significance of the geopotential model is its
ability, ) describe the gravitational field more accurately than a point mass mode! for the
Earth. For example, in the region of reentry below 120 kin, the first-order zonal
harmonic J2, may attain a magnitude approaching that of atmospheric drag [Ref. 18:p.
40].
c. Modeling Atmospheric Drag
As previously mentioned, the force that causes an artificial satellite's
orbit to decay is atmospheric drag. The atmospheric drag acceleration vector, in
equati-:- (4), acts in the direction opposite of the satellite's relative velocity vector and
is givt.u by [Ref. 17 :p. 258]
1 CDA ,2 6ad,, : 2 A
where
Ct, = satellite drag coefficient
A = aerodynamic effective cross-section area
m = satellite mass
po = local neutral density of the atmosphere
24
- ,
TESSERAL .
€+ -ZONAL SECTOR IAL
Figure 7: Earth Geopotential Model
[Ref. 171
25
VA = iV - w, x r I (satellite's airspeed)
v = unit vector in direction of IV - w. x )
w,: = 2w rad/day
V = satellite's inertial velocity vector
The ballistic coefficient, from equation (6), is defined as
B =_ COA (7)m
Atmospheric drag dissipates the satellite's energy which in turn causes
a decrease in the semi-major axis. Drag affects high eccentricity orbits by gradually
decreasing the apogee altitude, while maintaining a nearly constant perigee altitude,
resulting in circularizing the orbit as shown in Figure 8 [Ref. 17 :p. 258]. This
contraction will continue until the satellite begins the reentry phase. Under certain
simplifying assumptions, such as ignoring the rotation of the atmosphere, the analytical
results describing the change per one revolution in semi-major axis and eccentricity for
orbital decay arc given by [Ref. 19:p. 230]
3
A•, = - a P (8)
(1 -ecosE)2
CA +ecosAe = -- a] -- 'ccE. (cs E+e) dE (9)
where
a = semi-major axis
26
LUJ
IL
-AJ
[R'e . 1. .• .,
'J.of,
• • . o
g4
• |-
•?, '•.-. ,',.
,,-..•
Figure 83: Drag Effect On High Eccentricity Orbits(Ref. 1 71
27
e = eccentricity
E = eccentric anomaly
(1) Modeling Atmospheric Density. The Earth's atmosphere is
primarily composed of nitrogen and oxygen. Solar radiation affects the dynamic
properties of this medium by constantly changing the temperature, pressure, chemical
constituents, particulate presence and electrical properties [Ref. 20:p. 5]. The inability
to model the atmospheric neutral density is an integral part of the satellite reentry
problem. Neutral density is defined as the density of the neutrally charged constituents
of the atmosphere. The atmospheric density is not precisely known along the satellite
path because it varies with geographic location, solar and geomagnetic conditions,
altitude and time [Ref. 17 :p. 257].
Atmospheric density models are categorized as either "theoretical,"
or "empirical" models. Empirical models describe the phenomena of the Earth's
atmosphere, based on a summary of observed data and are constructed independently of
the laws of physics. Theoretical models apply the laws of physics. Empirical and
theoretical models may have overlapping domains. For instance, data used to construct
a particular empirical model may be smoothed and/or extrapolated based on theoretical
considerations. Likewise, theoretical models must be compared to empirical data in
order to define empirical parameters such as average sea level values of pressure and
temperature and to establish assumption limits. [Ref. 21:pp. B-1,B-2] For example, the
well known Jacchia models and the global dynamic models are semi-theoretical and semi-
empirical in nature. The global dynamic models describe the physical and chemical
28
processes of the Earth's coupled thermosphere-ionosphere system. Solar radiation and
auroral input measurements are used in the physical model to predict the time-dependent
density response. [Ref. 22:p. 1]
A simplified analytical atmospheric model demonstrates the
relationships between temperature, altitude and density
P He() (10)
where
PO = sea level air dersity
z = altitude
H = atmospheric scale height (RTo / g)
T0 = atmospheric tcmperature at sea lcvcl
R = gas constant (air)
Equation (10) is a simple analytic relationship where the atmospheric density decreases
exponentially with altitude when gravity, temperature and chemical composition are
assumed to be constant at all altitudes. This model rep.:esents a rough approximation of
the atmospheric density. Additional physical properties and levels of sophistication can
be incorporated into this simplified model in order to better describe the a.ctual
atmosphere.
Empirical atmospheric density models use a variety of functions to
describe the atmosphere. Some of these functions are common to most models. The
29
following list provides a brief description of these common functions: [Ref. 21:pp. B-13--
17]
1. Altitude--density decreases with increasing altitude.
2. Early afternoon bulge--maximum daytime temperatures cause the density toincrease at satellite altitudes.
3. Average 10.7 cm (F10 .7) flux--the average solar power per unit area at afrequency of 2800 MHz (X = 10.7 cm) measured at various Farth locations. The10.7 cm flux is closely correlated to the extreme ultra-violet (EUV) radiationwhich heats up the upper atmosphere and is used as an indicator of the EUVflux, since EUV flux is absorbed at higher altitudes and is converted to heat.
4. Daily 10.7 cm flux--accounts for the rapidly changing values of the F10.7
measurement.
5. Geomagnetic index (A.)--related to the activity of charged particles. Usuallymodeled as a correction to the atmospheric temperature.
6. Winds--speeds up to 300 mis have been observed in dite uppet ati..osphiler.Wind can significantly change the drag experienced by the satellite since dragis proportional to the square of the velocity with respect to the surrounding air.
Numerous empirical atmospheric density models have been
developed since the launch of Sputnik I [Refs. 23-31]. Early models such as the Jacchia
70 and 71 were derived from the analysis of satellite drag. These models identified the
upper atmosphere as dependent on solar flux, geomagnetic index, diurnal, monthly and
seasonal variations. A later model, the Jacchia 77 model incorporated composition data
of nitrogen (N) and mono-atomic oxygen (0) which was observed from satellite mass
spectrometers. The mass spectrometer and incoherent backscatter (MSIS) model was
30
constructed using composition data derived from satellite mass spectrometers,
accelerometers and ground based incoherent backscatter measurements.
Several comparison studies have been conducted with various
empirical atmospheric density models to determine their overall accuracy and efficiency.
These studies indicate the following limitations and deficiencies of empirical models
[Refs. 21, 32-34]1
1. Accuracies of models have remained relatively unchanged for the past twodecades.
2. Statistical analysis of measured satellite accelerometer density data as comparedwith atmospheric density model mean values and standard deviations, indicatemean value accuracies of approximately ± 10% with standard deviations ofapproximately ± 15%.
3. Some models are significantly less efficient in terms of computational time.
4. The Fo.-, cm measuren,eziL does not adequately represent the complex interactionbetween the EUV flux and the thermosphere.
5. The geomagnetic index, AP, or 3 hour K, does not necessarily represent thephysical mechanism that causes the variation in atmospheric density.
6. New model parameters such as the precipitation index (used as an indicator ofmagnetospheric activity), are needed to model the real physical variations in theatmosphere.
Additionally, our ability to predict the solar flux and the
geomagnetic field is usually difficult and unreliable at best. As a consequence, accurate
forecasting of atmospheric density into the future is limited. This affects the
31
determination of the predicted orbital decay rates because of the dynamic dependence on
the variations of the Fo0.7 and AP indices. [Ref. 35:p. 1]
Considerable progress has been made over the past decade in the
development of a dynamic atmospheric model of the coupled thermosphere-ionosphere
system. The National Center for Atmospheric Research (NCAR) has developed a model
as the thermosphere/ionosphere general circulation model (TIGCM). The pressure
coordinate primitive equations of the lower atmospheric meteorology are the foundation
of the TIGCM. TIGCM uses 20 minutes of CRAY-Y-MP 8/64 computer time per
simulated model day to compute the prognostic thermodynamic, eastward and northward
momentum equations and diagnostic equations of state and continuity. The model utilizes
a 50 lat-long grid with 24 constant pressure surfaces distributed from an altitude of 95
to 500 kr. Density data collected from a 200 km altitude satellite was compared with
the TIGCM calculated density and the results showed that the TIGCM calculations were
within ± 9-12% on a point by point basis along the satellite's track. A new version,
TIE-GCM, added an interactive dynamo model to calculate electrodynamic interactions
between the thermosphere and the ionosphere. Results indicate this version model is able
to provide improved determination of thermospheric density, especially during disturbed
geomagnetic conditions. [Ref. 22:pp. 1-6] An operational version of this model, using
a vector spherical harmonic (VStt) technique is under development at the University of
Michigan. With further development, this dynamic atmospheric model may be able to
forecast global atmospheric density values with errors less than 10%. [Ref. 36]
32
(2) Modeling Aerodynamic Coefficients. The aerodynamic coefficients,
depend upon the following factors:
1. The shape and dimensions of the vehicle
2. The vehicle orientation to the on-coming air flow (angle of attack)
3. The temperature and composition of the neutral atmosphere
4. The gas-surface interaction phenomenon
The flow phenomena encountered around complex shapes further complicates and varies
this parameter in the aerodynamic regimes: free molecular flow, transitional flow, and
continuum flow [Ref. 37 :p. 5]. A CD of 2.2 is typically used as a constant value in the
free molecular flow region above 120 kmn. However, CD is a function of angle of attack,
shape and flow rcgime. Therefore, knowing the vehicle's motion about its center of mass
is a critical factor in determining the aerodynamic forces acting on the vehicle. These
forces dictate the orbit decay rate, reentry and impact location. C, can vary from 2 in
the free molecular flow regime to a value much less than 1 in the continuum flow
regime.
(3) Rarified Gas Dynamics. The pressure distribution and subsequent
forces imparted by the near flowfield on a reentry body determines the forces and
moments acting on the body via the aerodynamic coefficients [Ref. 20:p. 203]. There
are basically five flow regimes which have distinguishable characteristics. These five
flow regimes may be given quantitative definition using the Knudsen number (Kn), T•e
Knudsen number is defined as the ratio of molecular mean free path to a characteristic
33
vehicle dimension, usually nose radius. The Knudsen number is indexed according to
its reference mean free path, either after collision or in the oncoming flow.
The boundary layer may be characterized by the Reynolds number
(Re). Flow is said to be laminar when the viscous forces are sufficiently large to damp
out oscillations caused by the dynamic forces. A low Reynolds number is characteristic
of laminar flow. Turbulent flow is said to occur when the dynamic forces overcome the
viscous forces, there is random mixing of particles and largc momentum exchanges
between fluid particles. When flow over a solid body reaches a critical Reynolds number
the initial laminar flow transitions to turbulent flow. Turbulent flow is a critical factor
in reentry since there is much more energy near the vehicle's surface than in laminar
flow conditions. In turn, under turbulent flow conditions, more heat is transferred to the
surfa( e of the vehicle. The five flow regimes are shown graphically in Figure 9 and
described as follows: [Ref. 20:pp. 203-206]
1. Free Molecule Regime--this region is where the molecular mean free path (X)is relatively large compared to a charactefistic vehicle dimension such as noseradius. When molecules collide with a boundary layer they attain the state ofthat boundary after a single collision. This flow regime is characteristic of theuppermost portion of the atmosphere.
2. Near Free Molecular Flow--frequently referred to as the "slip region" becausegas molecules will acquire the momentum of the moving boundary only afterseveral collisions. If, on the average, a molecule fails to acquire the momenzumof the moving boundary after a single collision, then it is said to lack"accommodation." This lack of accommodation means that the temperature isa nearly discontinuous function of distance away from the moving surface.
34
o z Ii
a. W .. 4 0 0
-J~ go~ 0 ~ 2 wu0 c C'4
cc < -4
LU,
OILj
00
-' U. I
I- j
-U-
[Ref. 20
LL cc35
3. Tiansition Regime--little is known about aerodynamic quantities such as lift,drag or heat transfer. This region is not treated very well analytically.
4. Viscous Merged Layer Regime-.-essentially that region consisting of dynamicallycoupled shock-wave, boundary-layer interactions. The presence of a boundarylayer on the wall alters the boundary conditions for the shock wave,simuh.lleously, the large pressure gradients due to the shock wave strongly alterthe boundary-layer flow. Neither the shock wave nor the boundary layLr maybe treated as discontinuities. This is the region from approximately 110 km to75 km; this is where the "initial pitch over" occurs and reentry "starts."
5. Continuous Regime--the region where classical fluid mechanics of high Reynoldsnumber applies, here the shock wave and boundary layers are again treated asdiscontinuities. Often this regime is subdivided into four categories (subsonic,transonic, supersonic, hypersonic) with lines of demarcation established byMach numbers.
(4) Vehicle Profile Area. The profile area, or aerodynamically
effective cross-section of the vehicle, is the area presented to the oncoming flow of the
atmosphere and is a function of the vehicle's attitude and configuration. A spherical
satellite maintains a constant profile area. More complex vehicles that have various
design shapes and deployed solar panels can have highly variable areas The ability to
model the area depends on the known dimensions and orientation of the vehicle. Since
an uncontrolled satellite reentry is characterized by a critical system failure or the end
of its opceational lifetime, communications with the vehicle may have been lost. Uader
this condition, information on the satellite's attitude will not be directly available [Ref.
37:p. 4]. Additionally, the vehicle will experience increasing aerodynamic and thermal
loads as the altitude decreases. These forces will act to change the vehicle configuration
by deformation and removal of structures (mass) [Ref. 5:p. iv].
36
(5) Vehicle Mass. Knowing the mass of a satellite accurately assumes
-a comprehensive knowledge of the vehicle. This information may be available for some
U.S. satellites, however, for foreign systems this may pose a problem. As mentioned
in the previous paragraph, aerodynamic and thermal loads can change the satellite mass.
The ability to model changes in mass also assumes an accurate knowledge of the
environmeni and how it affects the forces and subsequent motion of the vehicle about is
center of mass. In order to develop a model capable of predicting this motion, the
vehicle's moments of inertia must be known. [Ref. 37:p. 4]
(6) Vehicle Velocity. The relative velocity of the vehicle with respect
to the Earth's rotating atmosphere in equation (6) is
V, = I v-wooxrl P 011)
The maximum deceleration and heating rate experienced by a reentry body is a function
of velocity. As mentioned before, wind can change the vehicle's relative airspeed which
can affect the drag or lift experienced by the vehicle. Determination of the vehicle's
velocity using doppler range rate observation information is rtlatively accurate.
Uncertainties in the radial velocity may be as low as .166 m/s [Ref. 38:p. 138].
d. Modeling Aerody iamic L~ft
Lift causes the vehicle trajectory to follow a glide slope, skip path, or
perturbed ballistic flight path. During hypersonic flight conditions, lift is generated by
pressure forces on the lower surfaces at angles of attack caused by motion about the
37
center of mass. A perturbation from the nominal or zero lift flight path is the net result.
The force of aerodynamic lift is defined as [Ref. 3 9:p. 25]
L - 1.p CLAV2 (12)
where
CL = coefficient of lift
Reentry trajectories with lift -educe the thermal and struct-'ral loads on the vehicle.
During the final revolutions of orbit decay and reentry, lift is assumed
or modeled to be very small or zero [Ref. 1A:p. 2], [Ref. 39:p. 33]. However, an
uncontrolled reentering satellite may generate a significant lift vector depending on the
attitude, shape and motion of the vehicle about its center of mass. When this occurs, the
lift will not be distributed equally about the nominal flight path. This may result in a
deviation from the projected impact point. [Ref. 15:p. 47]
e. Reentry Equations of Motion
The reentry equations of motion can be derived by the mathematical
transformation of the second-order, nonlinear, vector differential equation (4). Several
analytical and semi-analytical theories describing a satellite's shallow reentry equations
of motion have been developed over the last thirty five years [Refs. 14,40,41,42].
With the advent of manned space exploration and recoverable probes, it becamenecessary to develop accurate theories of the entry phase, during which thealtitude, velocity, deceleration and the heating rate vary rapidly. [Ref. 41:p. 2]
In the derivations of these theories, several strong physical assumptions were made:
38
1. Spherical Symmetry--the Earth and its atmosphere are spherically symmetric.
2. Non-rotating Atmosphere--the rotation of the Earth's atmosphere which isapproximately equal to the angular velocity of the planet is neglected.
3. Exponential Atmosphere--the atmospheric density decreases exponentially withaltitude.
4. Gravitational Field--the gravitational field is assumed to be constant duringreentry at all altitudes.
5. Coordinate System--a non-rotating two-dimensional inertial coordinate systemwith the origin at the center of the Earth.
(1) Fundamental Equations of Entry Dynamics. The exact reentry
equations of motion derived by Loh, reference [40], are presented to show the basic
relationship of the vehicle's force vectors along the radial and normal direction to the
flight path in an inertial c,.ordinate system as shown in Figure 10 [Ref. 40:p. 19]. This
presentation serves as the foundation for the development of more sophisticated theories
to be presented in Chaptex III. Physical assumptions 1,3,4, and 5 were used in the
derivation of these equations.
The magnitude of the relative aerodynamic velocity,V1 , does not
equal the magnitude of the inertial velocity, V, due to the reentry body's trajectory
through the moving atmosphere. These velocities and their geometric relationships are
shown in Figure 11 [Ref. 4 0:p. 18] and are related by the following equation
Vr= V '12((1 - cos Yr. Cs a') - c Os Yr cos (13)
39
r1 Orog 7
e # Littc .gq. of moving vehicle
Local horizontalInertial velocity
Center of the Earth
Figure 10: Inertial Coordinate System
[Ret. 40)
40
Local vertical
- East
VrV
IV
South
Figuro 11: Relationship Between Relative And Inertial Velocity[Ref. 40]
4!
where
Vr = 1519 cos 65 (ft/sec) (velocity of the Earth's surface at a specific latitude)
6 = latitude angle
,y = 0 (flight path angle to the local horizon from Figure 10)
The force componr.ats from Figure 10 are defined as
S= -Dcos + +Lsin 4)-mgsin m A md(Vcos i)) (14)Ft
S= Lcos + +Dsin 4-mg cos = (vsin) (15)
dt
where
V V, (relative velocity)
By resolving equations (14) and (15) in the velocity direction, replacing L and D with
equations (12) and (6) respectively, and by further rearrangement of the terms and
simplification, according to Loh, the exact equations of motion are
dcosy + CILT (16)
42
where
radius of the Earth
= inverse atmospheric scale height (Mg/RT)
M mean molecular weight
The exact equations of motion of entry dynamics cannot be solved
analytically, however, solutions can be found using numerical methods. First-order
approximate analytical solutions have been developed by restricting the equations to
limited regions of application. [Ref. 4 3 :p. 25] The ability to accurately model and solve
these equations depends on the knowledge of the initial conditions: the initial position,
and velocity, the vehicle's area and mass, the neutral atmospheric density, and the
aerodynamic coefficients.
2. Modeling Breakup
a. Reentry Body Structural Mechanics
A satellite will experience structural and thermal loads during reentry into
the Earth's atmosphere. The structural response of the body during reentry may depend
on the following factors: [Ref. 44]
1. Static load effects
2. Dynamic load effects
3. Thermal load effects
43
The coupled effect of these factors may determine when the body will experience
structural failure.
b. Modeling Reentry Heating Effects
The reentry process is essentially that region of flight where the vehicle's
velocity into the atmosphere is reduced and its kinetic energy is converted into thermal
energy in the surrounuing medium. Since breakup is determined, in large part, by when
the outer surface reaches its melting point, reentry heating directly affects survivability.
This will be discussed in detail in Chapter III. T he conversion fraction of kinetic energy
to heat energy is a function of the satellite's shape, velocity, and altitude. At very high
altitudes (free-molecule flow region) the heat energy is developed almost entirely at the
surface of the vehicle and up to one-half of the lost kinetic energy may be converted into
heat in the vehicle body. [Ref. 43:p. 191] At low altitudes (continuum flow region) the
heat energy will appear in the area between the shock wave and the body. This heat is
transferred from the hot gas to the vehicle by conduction and convection through the
viscous boundary layer which is adjacent to the surface of the vehicle. Radiant heating
of the vehicle from the hot gas also occurs and this contribution to the surface heating
is dependent upon:
1. Bluntness of the vehicle's leading edge
2. Excess orbital velocity of the vehicle
44
Excess orbital velocity is defined as an appreciably higher velocity than the circular
orbital velocity for a given altitude. [Ref. 4 3 :p. 192]
The rate at which the vehicle heats is noi exclusively dependent upon this
energy conversion fraction, it is also dependent upon the rate of kinetic energy loss by
the vehicle. For the case of natural orbital decay, reentry is at small flight-path angles
and the deceleration is very slow in the upper atmosphere. The surface heating rate is
relatively low despite the high conversion fraction in this instance; therefore, the
dominating factor is the low rate of kinetic energy loss by the vehicle. Conversely, at
steep reentry angles, where deceleration occurs rapidly, the surface heating rate is high
although the energy conversion fraction is low.
The total heat input into the reentry vehicle depends upon the time of heating
as well as the heating rate. If the energy conversion rate were constant, the total heat
input would simply be a fixed fraction of the initial energy and the type of reentry would
not be of significance.
Three aspects of the aerodynamic heating process are significant, namely:
[Ref. 40:p. 181]
1. The total heat input, Q.
2. The time rate and maximum time rat2 of local stagnation region heat input perunit area (dH/dt) and (dHMdt)•,.
3. The time rate and maximum time rate of average heat input per unit area(dtt,,/dt) and (dHIdt),,•,.
45
The time rate of average heat transfer per unit area is given by equation (18) below. The
average heat transfer is simply the total heat input divided by the time of input. Overall,
reentry vehicle structural integrity is a function of the average heat input. [Ref. 40:p.
181] The time rate of local stagnation region heat input per unit area is given by
equation (19). Local structural integrity is a function of local stagnation region heat
input or the generation of local "hot spots." [Ref. 40:p. 182] The total heat input is
given by equation (20).
(1) Reentry Heat Input. From the three previously mentioned areas of
concern, the following generalized aerodynamic heating equations are given [Ref. 4 0:pp.
183-184]
dFI• l dO 1 (18)
dt Sdt 4
H2 -K[ V (19)= Kn Pr -V3 = KI I V3dt (f
jdId C,/ SPV( C;ýS V2dp (20)
Q = d7t -4 gsinV
where
H., = average convetive heat transferred per unit area, ft-lb/ft2
H. = local stagnation region heat input per unit area
Q = convective heat transferred, ft-lb
and
46
- -I (21)c; =P, [c1 f.|)#[•. CI]S
CF'- equivalent skin-friction coefficient
V = velocity, ft/sec
p atmospheric density, slugs/ft3
Kn = Knudsen number = X/Re
Re = Reynolds number
X molecular mean free path
A = coefficient of viscosity, lb-sec/ft2
Pr Prandtl number, subscript e indicates "entry" value
-y =ratio of specific heats
o =nose or leading edge radius, ft
K' constant 6.8 to 15 x 10l
Klý (22)Pr,
S = surface area, ft2
0 = flight path angle, positive for descent
CF, = coefficient of friction (local conditions)
CP = specific heat at constant pressure
47
Cil = specific heat (local conditions)
The above equations make the following simplifying assumptions: [Ref. 40:p. 18!j.
1. Radiative heat transfer from the surface generally does not appreciably influenceconvective heat transfer to a vehicle; therefore, it is disregarded.
2. Effects of gaseous imperfections may be neglected.
3. Shock-wave boundary-layer interactions may be neglected.
4. Prandtl number is constant.
5. Reynolds analogy is applicable.
If the heat transferred to the reentry vehicle is expressed as a
fraction of the total kinetic energy, then [Ref. 20:p. 138]
KE -m.V (23)
where
m = mass of reentry vehicle
VE = reentry velocity
Q = 2Q (24)mVE
where
Q = fraction of heat transferred to the reentry vehicle
Q = total heat input
48
This may be rewritten as
Q [2 !] [li_. -2] (2-)
where
CF = coefficient of friction
CD = coefficient of drag
S = reference area (usually nose tip)
Sw = wetted surface area
and where the trajectory parameter is defined by [Ref. 20:p. 138]
POC SH _ pogH (26)2 msin-yF 2I•sin'yE
when
"-YE = reentry flight path angle
At small flight path angles, such as reentry from orbit decay, aE will be very large,
therefore, equation (25) reduces to
1 F sw1 (27)
49
It is now recognizable that the fraction of the initial kinetic energy, transferred to the
reentry vehicle by convective heating, is one half of the ratio of the friction drag to total
drag. [Ref. 2 0:p. 138]
In the case of ballistic reentry at small angles of reentry, both the
maximum heating rate and total heat load increase as the effective mas3-area ratio
(m/CDA) or ballistic coefficient increases [Ref. 40:p. 195].
Assuming a constant ballistic coefficient throughout the reentry process
yields a different relationship [Ref. 40:p. 198]
Q. MA (28)
(dH(29)
Q 1 (30)
at) '-10
(dlsL 1 (1
Therefore, reentry at small angles of inclination reduces the maximum heating rate of the
vehicle, however, the total heat load is increased. [Ref. 40:p. 198]
50
(2) Reentry Heating Rate. If the heating rate per unit time is defined
[Ref. 20:p. 135]
d = dq (32)dt
then the total heat-transfer rate may be written [Ref. 2 0:p. 136]
f- s (33)
After some simplification it is possible to write [Ref. 2 0:p. 137]
CF SW W (34)
4
this assumes
fCs CFs' (35)
It is now possible to define dq/dt as the average rate of change of heat transfer per unit
area
(36)4
It can be shown that dq/dt is a maximum when pV3 is a maximum. [Ref. 20:p. 137]
(3) Ablation. As previously discussed, the heating experienced by
reentry at small flight path angles is different from that of reentry at large flight path
angles. The flight duration is much longer for the first case and even with a lower
maxi,,,dm heating rate, the total heat input exceeds that of the larger reentry angle. The
5!
predominant cooling mechanism of uncontrolled reentry of vehicles not designed to
survive reentry is ablation. Ablation is the term which generally applies when:
... there is a removal of material (and an associated removal of heat) caused byaerodynamic heating, and therefore embraces, melting, sublimation, melting andsubsequent vaporization of the liquid film, burning and depolymerization. [Ref.15:p. 198]
Ablative materials are measured as "effective" depending on their
capacity to dispose of heat (latent heat) by convection in the liquid film, and by
convection in gaseous form in the boundary layer. Previous work has shown that when
a material experiences a large percentage of total mass loss as vaporization, it is a more
effective ablative material. Sublimation is the process whereby all the mass loss
undergoes vaporization, there is no liquid film, and therefore is an excellent method for
removing large amounts of heat. For these reasons, materials which undergo sublimation
at reasonably high temperatures are generally more efficient at removing heat and thereby
reducing the thermal load on the reentry vehicle. [Ref. 40:p. 204]
Radiative cooling is another mechanism by which heat is removed
from the reentry vehicle during reentry. Radiative cooling and ablation combined are an
effective pair in balancing the heating effects of reentry for a lifting body. For a non-
lifting body radiative cooling is inefficient.
C. GENERAL DESCRIPTION OF THE REENTRY PHASE
A decaying satellite possesses a large amount of kinetic energy due to its velocity
and potential energy because of its altitude above the Earth's surface. As the satellite
encounters the atmosphere, a shock wave forms ahead of the vehicle which heats up the
52
atmosphere surrounding the vehicle. This enveloping layer of incandescent atmosphere
causes the vehicle's temperature or thermal load to continually increase as it penetrates
into an increasingly denser atmosphere. During this phase, the velocity continues to
decrease as the kinetic energy is converted into heat through the atmospheric drag. If
all the satellite's energy were converted into heat and contained within the vehicle, then
there would be more than enough energy to vaporize the vehicle. However, this is not
the case. A large part of the total energy is diverted away from the vehicle by two
processes. The first process unloads a major fraction of the heat into the atmosphere by
a strong shock wave mechanism. The second process involves the radiation of heat away
"rom the hot surfaces. [Ref. 4 3 :pp. 1-2]
The vehicle also experiences structural loads, which are a combination of
aerodynamic and inertial ioads [Ref. 4.:p. 51. Atmosphenc drag forces usually cause
a reduction in the vehicle's kinetic energy. Centrifugal and lift forces cause accelerations
normal to the direction of the motion.
Aerodynamic lift and drag forces vary directly with the square of the vehicle's
velocity, V2, and with the atmospheric density, p. Deceleration of th. vehicle is a
product of two quantities, density and velocity. As the satellite penetrates further into
the atmosphere, the density increases rapidly resulting in a corresponding decrease in
velocity due to drag. Initially, deceleration increases as shown in Figure 12 [Ref. 4 3 :p.
7]. However, at some altitude the velocity begins to decrease at a faster .-ate than the
increasing density, which results hi a maximum deceleration. Additionally, the
maximum heating rate, - pVJ, occurs at an altitude somewhat higher than the maximum
53
I.2
4-
>
I-
4-
03
C.)
Figure 12: Chan~ges During Atmospheric Reentry[Ref. 43]
54
deceleration [Ref. 20:p. 98], [Ref. 4 3 :pp. 6-7]. A maximum deceleration of up to 8
g's, as shown in Figure 13, [Ref. 14.p. 30], occurs during this phase [Ref. 3 9 :p. 36].
When the satellite's skin temperature and structural load become sufficiently high,
the vehicle will start burning and breaking up. Solar panels and other Projections such
as antennas will separate at the earliest stage, while heavier pieces will breakup at lower
altitudes. Based on predicted and observed data, satellite breakup commences at an
altitude between approximately 75-120 km [Ref. 10:p. 5], [Ref. 46:p. 4], [Ref. 47:p.
39]. The resulting debris from the bretakup will impact the Earth's surface provided it
survives the reentry heating process.
D. CURRENT REENTRY THEORIES
PredictinP reentry time and impacet location reflies iinnn nh-,Frvatinnq] dt1qt nf the-
re~entry vehicle. Based upon observed (measured) position and velocity, over thle orbital
path and ideally equally distributed over that path, an algorithm is used to calculate an
elliptical orbit which best fits the observational data, If the algorithm attempts to model
thle physical reentry process, it will be referred to as a physical model by the authors.
Another type of model to be discussed is the King-Hecle or mean motion type of model
which neglects certain physical asplects of the reentry process and focuses more on thle
observational data. The algor-ithm which "1fits" the observational data to an elliptical
orbit may also be used to propagate or predict future orbital locations as a fuinction of
time. These types of algoi ithrns are commonly referred to as propagators.
55
Present onaysis
- Approximatehuman tolerornce 4 ric
12 (ropid onset of g) -io,
(Allen- Eggers)
"u, 26,000 f ps,Ii __!.! _aJ7
2 5 4 6
Inioil angle,-4, deg
Figure 13: Maximum Deceleration vs Initial Flight Path Angle[Ref. 141
56
1. Physical Modeling Theory
The current propagator in the United States, for reentry prediction, is the
special perturbations (SP) model. This model is maintained by the Air Force Space
Command and is the standard for reentry prediction in the United States.
Orbital periods of less than 87.5 minutes are defined by the Space.
Surveillance Center (SSC), Cheyenne Mountain Air Force Base, Colorado, as decaying
orbits [Ref. 48:p. 3]. The SP model uses numerical methods to incorporate zonal,
sectoral and tesseral orbital perturbations in the calculation of decaying orbit reentry
predictions. Gravitational perturbations are modeled by mapping the Earth into small
grids, which allows for enhanced resolution of the geopotential. Third body gravitational
effects, sun, moon, and planets, may also be modeled. Third body gravitational effects
are used primarily in the propagation of highly eccentric orbits with large apogee
altitudes. The atmosphere is modeled using the Jacchia-Nicolet model which considers
the following: [Ref. 48:p. 9]
1. Diurnal bulge
2. Solar activity
3. Geomagnetic activity
4. Semiannual variation
A techniiue known as differential corrections is used to "fit" the observations
of the rcentry vehicle into the best orbit. The differential corrections are a mat.iematical
means of determining a single orbit path (ellipse) consistent with the observed data (the
57
only "known" information in the absence of active telemetry data). The need for the use
of differential corrections arises from the fact that the model used to propagate the orbit
into the future has inherent deficiencies. These deficiencies are then reflected as errors
in the prediction of where the vehicle is supposed to be in its orbit as compared to where
it is observed to be. Through a series of fitting observations to an orbit and updating the
predicted orbital path, the reentry of the vehicle is calculated as a "time" when the
altitude will reach a specified minimum value. This lower limit value of altitude is a
function of the atmospheric density model. The impact dispersion area or footprint is
calculated as ± 15 minutes of that reentry time. The sub-satellite ground trace is used
to describe the impact area on the Earth's surface. Table 2 shows the format of
observational data as used in the differential corrections process [Ref. 49]. Figure 14
shows th~e differential cmrr'ctinn3s display and Figure 15 shows the Trackina and Impact
Prediction (TIP) display used at Cheyenne Mountain AFB. [Ref. 49]
It now becomes obvious that there are two extremely significant issues at
hand:
1. The + 15 minute window equates to approximately 113 of one completerevolution of the reentry vehicle's orbit in the decay phase.
2. The lack of observational data in the decay phase may significantly bias thepredicted reentry time.
58
Table II: OBSERVATIONAL DATA FORMAT[Ref. 49] •=
--------------------- *II--e-----
* I I i .
- III
p . ,q .0 - °; ' c .n '
o1 a n , o ro n a a a , ci , ,., 0 0 ;
... ..-.!........ .... .. •....SII II
• .I .I I . . . . I . . . . .-- JINN *Z.% 2N: iI aI! --C- .--N N , IaII* I-I I . l iN Sl- , N 0
":I : l -l - - 0I a'
* • • ,4 6f 6 4 C 05' • N 4' - ' 4
* I j0 I'I I'I I'I I'II'i I'i I"
S I Ii
-)P # PI :4 -l_.1
- S 0 01-
LL- c4i
El- r-,.
Wi LfL
Cr LO
W LiJr.3 LiLiJ
0 * 0
LJ. wiC %.. %- .Z~ ~~~ Li CCd iLn
X.. Z L.J CD N 2 L LDt L7 O ~ gh~~-ckL ' G i-z Ne-m0 w I., "I If Li Idr
==W.- I- CN 'aNiz;ýCmoz AU2- I__ w .. J- i z Qm -
o- Nm -n 01ee M W) C4 P" c C 1
MJ1 Li a.LJ Vci nML C1L IM ýE
CU IaLr n ICI-; k -a O M(n-C ILIL ' 3 Ir"~ CE1WI = r -V1C m m=LirI C% rMr
ciL.J r4NXW..~- Cr Z bN r C ' M" r ý 4(L'i CLa, Lnr 0n-CC O~ý LL 3
- r.f 0 OC4MC3C ýLl M0
Li
N- L. aL-' Q,100
CNZL LZ O --- i.~LA I- i L - - - 0
OsZ 00 4"i -= ..C. Lnrn...1.-~~~~ ~~~ Li Z O0Lni ci 0 0L iL
CL Wi a- a - w > o i n a
---- =i =i.~i .- O j~ a Li Z Z: 2
0~~~~ 4z Li I----.I- I--01-. L-L.J Z - LZ
LiU -LLJj a miOC 0 :
i i Lý ci I I i i iUJ W- Oa 3 " Li0L&JLiJ0:t.- - L ,L L .
Li cL I-C Lk1 X - .Lj Z X ,zC
Li WWL5i ~ w M -Liw == m
Figur 14: Difreta Corcin Display1CD=w -I- www[R of 49]- LJ LI qL -C)6 A -
L60
LaJ LIJl- j- "I, / N
ULO
W) LO~
L&J
I.. LAJ - L
Fý I. LU U LLi
EMf I.- C e e
L6 - I.- = 0
( - ii 4- z
I.--
Len
LALO
NiJ MJ(
CD Le- 'Nnme I =r C~ll-7w , k
Mn
UU
LU a.
P~en$5-,
Figur 15 Trckn An matPrdcin(TP ipa[Rf 49] .J=
61%o -
The accuracy of the reentry prediction is directly limited by the ability to
cbserve the reentry process as well as indirectly by the inherent deficiencies in the
models used to represent the physical reentry process.
2. Mean Motion Theory
In the early development of his mcan motion theory of satellite lifetime
prediction, King-Hele defined orbital lifetime as the time remaining until the eccentricity
of the orbit reached zero [Refs. 19, 50-55]. This was a prediction of reentry based on
the circularization, or contraction of the orbit, due to atmospheric drag. A principal
assumption in this theory was that perigee height remained constant. When oscillations
in perigee height were considered (which occurs when an oblate atmosphere is modeled),
the theory Lad to be revised in order to take into account a zero eccentricity prior to
reentry [Ref. 54]. The definition of orbital lifetime was then redefined in terms of mean
motion, n, where a value of n (chosen by the user) represents an "end value" of a
circular height. For example, a value of n equal to 16.5 rev/day would corrcpond to
a circular orbit altitude of 150 km.
In the prediction of orbital lifetime, the observed rate of change of orbital
period, dT/dt, was the entering argument (where T is the orbital period) [Refs. 19, 50-
53]. This was equivalent to using the rate of change of mean motion, dn/dt, since the
two are related by [Ref. 54:p. 5]
n = -1 (37)7,
62
where
n = rev/day
T = days
In its simplest form, orbital lifetime is now expressed in terms of the rate of change of
mean motion as
L= 2 . (38)
where
L = lifetime in days
dn/dt = rev/day2
Q = function of e, n and H (density scale height)
arid expanded by [Ref. 54:p. 8]
Q 3enI°(z)[1 -0. 1j] (39)
where
10,11 = Bessel functions of the first kind, of degree 0 and 1, with argument z
J = 0.3eO - 0.025
z = ae/H
For the circular orbit, the lifetime may be written [Ref. 54:p. 9]
63
L =- 3H-Hn(I -e -) (40)
2aii
where
H.4 = scale height H km below initial altitude
h = decrease in altitude until reentry (120 kin)
In this form, density scale height is the only parameter which is not directly
derived from orbital data (observations). It can be shown that H is less sensitive to solar
activity than a neutral density atmosphere model, therefore, uncertainties in solar activity
will have less influence on the orbital lifetime prediction [Ref. 54:pp. 9-10]. Perhaps
more importantly, the mean motion technique does not require any knowledge of:
1. Satellite attitude
2. Satellite size
3. Satellite shape
4. Satellite mass
5. Aerodynamic characteristics
These factors are all accounted for in the observation of mean motion and rate
of change of mean motion. In summary, the mean motion theory is dependent upon five
parameters: [Ref. 54 :p. 12]
64
1. mean motion (n)
2. eccentricity (e)
3. daily solar activity index (Fo07)
LT,. average solar activity index (F1o. 7)
5. geomagnetic index (AP)
The last three terms above are incorporated into the calculation of at.nospheric scle
height.
65
IL
III. FORMULATIONS AND SOLUTIONS OF REENTRY
This chapter describes tht; "state-of-the-art" formulations and solutions (analytical,
semi-analytical, and numerical) of decay-induced reentry as determined from the
literature survey. Fundamental physical processes that were introduced in Chapter II,
including reentry equations of motion, rarefied gas dynamics, reentry heating and reentry
body breakup are presented.
A. ANALYTICAL REENTRY EQUATIONS OF MOTION
1. Chapman's Approximate Analytical Entry Equations of Motion
Chapman derived PA nnnline-snr rond-order differential enuation by reducing
equations (16) and (17) and by introducing a set of completely nondimensionalized
variables [Ref. 14:pp. 3-14]. In the development of this equation, physical assumptions
1 through 3 from Chapter 11 were used. Additionally, two mathematical assumptions
were used:
1. The fractional change from the center of the planet, dr/r, in a given iwementof time is much smaller as compared to the fractional change in velocity, d(Vcos -y)/V cos-y, given by the mathematical expression I d(V cos .y)/V cos y ISI dr'r I .
2. The flight path angle, -y, related to the local horizon direction for lifting vehiclesis sufficiently small such that the lift component in the horizontal direction issmall is compared to the drag component, given by the mathematicalexpression, I ;h- I (L/D) tan -y I .
66
It is erroneous o think that these assumptions will restrict Chapman's analysis toentry trajectories with small flight path angles and small lift-to-drag ratios as manyauthors have believed. On the contrary, these assumptions applied simultaneously,constitute a well balanced set of hypotheses and make Chapman's theory applicableto a large family of entry trajectories. [Ref. 42:p. 1791
As previously mentioned, Chapman introduced a set of dimensionless
(independent and dependent) variables. The independent variable is
S = Cos , _U (41)
and the dependent variable is
2mF3CrA T-(42)
Chapnman's final derivation is given by
--- • COS4 ýorL COS3 (3 )
1. 2. 3. 4.
where 3 = Mg!RT and the numbers associated with the brackets in equation (43)
represent the following physical quantities:
1. Vertical acceleration.
2. Vertical component of the drag force.
3. Force of giavity minus the centrifugal force.
4. Lift force.
67
For the specific case of the shallow reentry of a satellite from a decaying
orbit where the flight path angle is very small, the Z function equation (43), can be
written in the following form
Z _ r~ _ 0-L (44)
du u u) i 7Z D
with the initial conditions of
.-- = I ,,, = 0 (45)
and where the corresponding boundary conditions tbr decaying orbits are
Z(1) = 0 , Z'(l) = 0 (46)
Integrating equation (44) by numerical techniques for each designated initial
velocity, initial flight path angle and L/D ratio generates a solution. The results of the
solution are applicable to any vehicle of arbitrary dimension, size, or mass. Figure 16
shows the solution graph for a non-lifting vehicle (L/D=0) [Ref. 14]. Figure 17 shows
the solution graph of various L/D values using the parameter [Ref. 14]
SL (47)
The ratio u/u, foi the ordinate from Figures 16 and 17 is equal to equation (41).
Additionally, the following engineering quantities of interest during the
reentry process can be calculated from the Z function:
68
14:A
24 .... ... 1:HN!Mvl. i I
717 7 ft-T
T. 1
.. .... Ai-.20
T 4' ý!:4 7-t
jr ... .... .... .... .. .... .... ...
ti w 4-v
44
.12M715ý ,"4 '14 4 1: 1 1: T--. I T171.71.1J: Mt
-:4
-H jjjejr
i: i%
7. R OT:
7 7 4 4-:_M
I T
.04 1 Y4Tt
q-1 w P; 4:;
0 .4Dimensionless ,aýfcnify, iii--IL
Figure 16: Z Function Solution Graph Fo.- UD=O[Ref. 14]
69
I-fi
~~I am .,
(~~)u;II N4IJ
Drnr i oji:ess NJ~c 3yA
Figure 17: Z Function SoluTionlal FvVaiu !'
[RefT 14
lit70 T
1. Deceleration
2. Descent angle
3. Range
4. Time
5. Density-velocity relationship
6. Dynamic pressure
7. Reynolds number
8. Heating rate
9. Total heat absorbed
2. Loh's Second Order Unified Solution of Entry Dynamics
Loh derived a general second-order solution of reentry mechanics that covers
the entire range of initial flight path angles and L/D ratios [Ref. 40:pp. 25-36). Figure
18 shows the entire range of the second-order solution as compared with several other
analytical approximate solutions available in 1963, including Chapman's approximate
analytical theory [Ref. 40:p. 26].
The second-order unified solution can be derived from equations (16) and
(17) by using approximations, a binomial series expansion, and integration techniques
for a small flight path angle,-',, and is given by the following equation
71
Loh's hposAle nEgr'
lo~eoeaie L/) sm o I U0 large positive (LID)smnoI I ond large 8f
L Lees' (Ting
I median 0f
Gozleyll (LAO)
Figure 18: Loh's Second-Order Range Of Applicability[Ref. 40]
72
(OR Xf [nR3] -xf,
Xif
[[I 3 oJ ~ 2 [J ['~] 1(48)where
e ((49)
Ro = radius of the Earth
f = condition at beginning of unpowered glide or ballistic entry
F,(-), In + -(Cy) +(CC-y)l+ (C,-,)] e c' .1 (50)
(C,'Y)3 (C1 'Y)2 1+1 (CI')- -
[720c 240C' 12oc0 12Cr ] 120C4 12C'J
C, Wo. I Cos -Y gP,° -1(5)---2 [ j V2 (5i)
For the specific case when L/D=0, equation (48) can be simplified and
rewritter. as
73
In g = [&A [i-J [cos Ts-c ] (52)MO OS -Cos (2
Figure 19 shows a comparison between Chapman's first-order solution and
Loh's second-order solution for a small L/D=0.1 [Ref. 40:p. 63]. Both solutions offer
about the same degree of accuracy.
However, the first order solution is limited by the conditions that (L/D) tan -y mustbe smaller than 1 and the initial angle of inclination 7y must be very small; thesecond order solution does not suffer these particular limitations. When L/D-=0and -yf - 0, where the first order solution was not available previously, the S'-condorder solution offers for the first time a satisfactory solution in this region.[Ref.40:pp. 50-51]
3. Yarosnevskii's Entry Theory
Yaroshevskii develc -d a semi-analytical reentry trajectory th,"ory which was
originally published iin a Soviet journal, Kosmicheskie lssledovaniya, in 1964 [Ref. 42:pp.
158-176]. [An English translation of this article was not available to the authors.]
Using some simplifying assumptions, he derived a nonlinear second-orderdifferential equation which can be integrated analytically by using seriesexpansions. To some extent, Yaroshevskii's theory is a special case of a moresophisticated theory developed by Chapman. [Ref. 42:p. 158]
In the development of the theory, physical assumptions 1 through 5 apply to
the basic reentry equations (16) and (17), from which the differential equation was
derived. Also, for a constant angle of attack, a, the drag coefficient, CD, and the lift
coefficient, CL, were assumed to be a function of the Mach number.
74
20 400
0
- Numericol integrotion to, ARDCatmosphere (Rubesin-Goodwin )
--- Chopmon's analysis1 * * Loh's analysis(second order solution)
PR ~ ~90014 280 /
12 240
f o 200 =u-4J 4--= -1
"A
L-0.'
S120 1 3-A .2 lbff--2
4 80_(•
2 40 - " ,,.
0 O A I ._ _ _ 0_
0 0.2 0.4 0.6 0.8 1.0V cos 0
.1-0Figure 19: Loh's Second-Order vs Chapman's First-Order[Rcf. 40]
75
Yaroshevskii used an independent variable, x, and a dependent variable, y,
defined as
xCD(1)dV (53)CD@'V)
and
Y CD(1A r0 (54)
2m j3
where
r0 radius of the planet
/ 1 55)Vgro
By defining the differential relations dx and dy, using equations (53) and (54) along with
the equations (16) and (17) and by eliminating the flight path angle, -y, the second-order,
nonlinear differential equation is given by
dyC J ) I - -(
d2Y _• V (x) (56)
CD c(1) y
Solutions for equation (56) can be obtained by numerical integration. When
C. and CL are independent of the Mach number, solutions can be obtained by integrating
a selected series expansion of equation (54), depending on the type of reentry trajectory.
76
For the specific case of the shallow reentry from orbit decay, where
CLiCD=0, equation (56) can be derived as
d2y _ e z- 1 (57)d~X2 y
with the initial conditions of
xi = 0 y(O) = 0 dy = 0 (58)
By taking into account a singularity at y=O, the series solution to equation (57) is
8 2 a~ + . (59)y = xF (ao+a 1x+a x2 +a3x3.... )
where
a. =1
a, = 1/6
a2 = 1/24
a3 = 47/4752
The coefficients ak can be calculated by the recurrence formula
2k k-I
2 - E (2m + 1) (2rn + 3) a,,. a k-,,
aA (k + 1)! 3-., 1 (60)
1 + (2k+ 1)(2k+3)3
As in the case of Chapman's theory, several engineering quantities of interest
during the reentry process can be calculated from equation (56):
77
1. Time of flight
2. Range
3. Deceleration
4. Heating rate
5. Heat absorbed
4. Universal Equations for Orbit Decay and Reentry
Longuski and Vinh derived a set of universal entry equations of motion fer
all regimes of atmospheric flight: from the free-molecule flow regime to the near free-
molecule flow regime where orbital motion is perturbed by air drag, through the
transition regime to the continuum flow regime where the dynamic phase of reentry
occurs, and to the point of impact on the planet's surface [Ref. 41].
Rigorous mathematical techniques, such as Poincare"s method of smallparameters: and Lagrange's expansion are applied to obtain a highly accurate,purely analytical theory for orbit contraction and ballistic entry into planetaryatmospheres. (Ref. 41:p.v]
Figures 20 and 21 [Ref. 41:pp. 16-17] describe the inertial coordinate system,
nomenclature and the aerodynamic forces of the equations of motio~l for a vehicle with
a CL/CD ratio, defined by the following equations [Ref. 41:p. 10]
dr Vsin (61)
dO - Vcos -cos (62)dt rcos4
78
I
-Mw ve
r r
Figure 20: Coordinate System And Nomenclature[Ref. 41]
79
Xd
- " VerhcoI PIane
Ujft-Orag A L o
"-% ,, ,
pianoPan
D ... D.r
0 .
Figure 21: Acrodynamic Force Diagram[Ref. 41]
80
d4, Vcos 'ysin , (63)dt r __-
dV _ pACDV- -gsin y (64)dr 2m
Vy pACLV2COG a v2Svd p~V'c o_(9 _ -)tCos Y (65)-
dt 2m r
-pACVCS a V2 cos t tan ,p (66)dt 2m cos Oy r
The exact universal equations of motion for entry trajectories for a vehicle
inside a rotating atmosphere can be derived from a transformation of equations (61)
through (66) using the modified Chapr,.?n variables
V2cos2 -' (67)U -
gr
Z pACD r (68)2m -3
and nondimensional independent variable
s = o cos-ydt (69)
The exact universal equations for entry trajectories are [Ref. 4 1:pp. 11-12]
81
dZ = I Op_ -d _ 1+ 1 d] Z tany (70)
du _ 2 10'rZu [ +f- cos cr tany + sin -y (71)Ws Cos- CD 2 v/r ZJ
d-y . 1OTz Cos U Cos'y 1cosý (72)
da = V~rZ since f l nL o (73)T tan i cos2 -Y CD
dil = F~ Zsi a s in[Ca (74)' s i Sfi COS2 y CD
di = _O- Z cosc[ sinco2 (75)Cos 2oELCD
where
0l = longitude of the ascending node of the osculating planc
i = angle between the ascending node and the position vector
= inclination of the orbit
or = bank angle
82
Aco~rding to the authors, the universal equations have three advan'- ges: [Ref.
41:p. 128]
1. They are exact.
2. They are free of any restrictive assumptions.
3. They contain the modified Chapman Z variable, which permits a singletrajectory solution for a specified initial velocity and flight path angle thatapplies to any vehicle of arbitrary area, mass, or CD.
As previously mentioned, equations (70) through (75) can be used during all
phases of aerodynamic flight and are most useful for analyzing the last few revolutions
and the reentry phase. The accuracy of the equations depends on the readjustment of
the value of the inverse atmospheric scale height, •, for each layer of the atmosphere.
[Ref. 41:p.15]
Fr ,hespcnific . ,fr t froofa circular orbit, where C. =0, Lo gUski
and Vinh derived a separate analytical theory from the exact universal equations for entry
trajectories, due to the fact that:
... it does not seem possible to have a single analytic solution which is uniformlyvalid for all values of initial flight path angl :s because of the nature of theproblem. In the case of atmospheric entry from circular orbit, the magnitude of'the flight path angle, initially zero, rapidly increases, approaching 900 as 'evelocity becomes small. On the other hand, for steep angle entry, the flight pau,-angle changes very little--of the order of tenths of a degree--as the nondimensionalvelocity decreases from unity to one tenth of the original value (between Mach 2and 3). [Ref. 41:p. 851
Under the condition of reentry from a circular orbit, equation (73) is equal
to one and equations (74) and (75) are equal to zero. By using this fact and the variable
83
V 2=V U (76)v = - - cO176)gr cos' -
equations (70) through (73) can be written as
dZ (77)
dv - v in [_ i- (78)TO, cosY2 V-Z v
dy = 1 ( 1 (79)
By dividing equations (77) and (79) by equation (78) tG form a new set of equatiosn, and
defining the following change of variables for substitution into the new set
Y = 2Z (80)
,t ý -4-r sin7 (81)
X = -1L v (82)
i1
r= 1 (83)
"ani then by expanding this new set of equations in one term, in c, and using Poincare"s
method of small parameters for integration of a system with the assumed solution form
84
4•o + C (84)
S= Yo + CY, (85)
two systems of two first-order differential equations are formed [Ref. 41:p. 89]
d =' o (86)
d-I'o e X.. (87)
dY YO
r 2d( 1 _ ex- L1o (2ex-1)-2 -Y (89)-X T'o- IT -÷ 7o]
Equations (86) and (87) can written in the following second-order differential equation
form
d 2Yo _ ex-1 (90)WX7 Yo
and with the initial conditions for the case of the shallow satellite econtry
Yo(O) = 0 , 'I,0(0) = 0 (91)
a series solution can be obtained in the following form
85
2 [1+ L+±r +1 [X]'+ 4 [X3 ,+20021 X]1 (92)YOý X L 3 -4J 9L4 4 T 605880 4J j
4o =3X7 [i5(X (X]247[x13 20021 X 14](9
9 4 j 18 4 198 1 T 652-40 4
Yaroshevskii's approach was used to find the first term of the series in equations (92)
and (93). The same approach can be applied to equations (88) and (89) to find a
solution.
Figures 22, 23, and 24 represent various solutions graphs for equations (92)
and (93). The dashed line indicates the exact numerical solution while the solid line
represents te analytical solution. Figure 22 shows the variauons of aeiodyniaiimic
deceleration, G's (g's = number x gravitational acceleration at a radial distance r), as
a function of the dimensionless velocity, v, at several initial flight path angles,-y,, for
equation (92) [Ref. 4 1:p. 120]. Figure 23 shows the variations of In ( Z / 4)--( ro- r)/H
-- drop in altitude, in units of scale height, as a function of (v) at several (n,), for
equation (92) [Ref. 41:p. 122]. Figure 24 shows the variation of the negative flight path
angle, --y as a function of (v) at severial (-yi), for equation (93) [Ref. 41:p. 121].
5. Attitude Dynamics of Uncontrolled Motion During Reentry
In the previous section several analytical theories were presented describilIg
the reentry equations of motion and their sohltions. Strong physical assumptions were
made in the derivations of these theories in order to describe the trajectory of the body's
86
, •'i = -I0o
0 -••
24
20.0 . 00 4.00.01 0
88
0.0 020 04.06 .8 0
mBV
Fiue2:Vrain f| sTh odmninlVlct v
[Ref. "41.°J\
,87
20---
12--
20
0
zz.
.4
--4
z ~ ~~~- 00°==' .1
0.00 0.20 0.40 0.60 0.80 1.00V
Figure 23: Variations Of ln(Z/4) vs The Nondimensional Velocity (v)
IRef. 41]
88
14
121
<~ .7
6 S 0
20 -- 0 -=
0.00 0.20 0.40 0.60 0.80 1.00
Figure 24: Variations Off G vs The Nondimensional Velocity (v)[Ref. 4J11
89
---- I' --- --- --
"center of mass or point mass. Specifically, Chapman's and Longuski and Vinh's theories
used tne Z variable which permits a single trajectory solution for a specified initial
velocity and flight path angle that applies to any vehicle of arbitrary area, mass, or CD.
The effect of the uncontrolled motion of a body about its center of mass on the reentry
trajectory was not investigated in these theories. This section will Lxamine three
analytical investigations presented by two Russian authors in this specific area.
Duzmak, in 1970, presented the first systematized explanation on the problem
of the attitude dynami'cs of uncontrolled reentry body motion. This problem was the least'
developed in comparison to center of mass or point mass trajectories and aerodynamic
heating [Ref. 5 6 :p. 2]
Unifying papers on the dynamics of the motion relative to the center of mass havenot appeared up to the present time. [Ref. 56:p. 5]
The primary focus was the investigation and derivation of the relationship
between a reentry body's parameters outside the atmosphere with the body's parameters
in the dense layer of the atmosphere. Changes in the state of a reentry body's motion
relative to its center of mass during zhe motion along its trajectory were also investigated.
An asymptotic approach was used on the equations of motion to solve the problem,
specifically for the cases where the characteristic time of the entry body motion relative
to its center of mass is much less than the characteristic time of motion of the center of
mass. Additionally, a refined asymptotic method that has a significantly wider range
of applicability was developed foi those cases in which the above condition was not
fulfilled. This method is based on the coupling of the numerical and asymptotic
90
solutions of the equations of motion. Two-dimensional motion without restrictions on
the shape of the body and three-dimensional motion of a body that possessed
aerodynamic and dynamic axial symmetry were assumed in this investigation. [Ref.
5 6 :pp. 1-3]
During the orbital or exoatmospheric phase, the external moments that
produce the reentry body's perturbed motion about its center of mass are determined by
the laws of motion of a rigid body as described by Euler. The magnitude and direction
of the initial angular moment vector, H, determines th- state or nature of this motion.
For instance, if initial angular moment vector during the orbital phase produces two-
dimensional mriotion, ti the body rotates at a constant angular velocity, W, about its Z
or transverse axis. [Rt:;. o:pp. 5-6] Figure 25 is an attitude dynamics coordinate system
that shows the direction of the Z-2xis relative to the body's orbit [Ref. 57:p. 113].
Regardless of the nature of the motion, t,,e entry body's angle of attack, u, uponapproaching the atmosphere can have absolutely any value at all ... Thus anycomplete solution of the problem of atmospheric entry can be obtained upon thenecessary condition of the absence of any restrictions on the size of the angle ofattack. One can say that the presence of large angle of attack is one of the maindistinctive features of the problem of atmospheric entry. [Ref. 56:p. 6]
As a body with perturbed motion relative to its center of mass enters the
reentry or atmospheric phase, it begins to experience a stabilizing effect that is
proportional to the incroase in atmosphefc density. This stabilizing effect is a property
of dynamical systems with variable parameters where if the stiffness of the system
increases during the perturbed motion, then the vibrations damprcn out. This dampening
effect was described, for the case of angularly misaligned missiles during reentry, by
91
ZO•
NorthEarth
1L1 EarthOrbit
YO
Vernalequinox
h~icky o~Y
Figure 25: Attitude Dynamics Coordinate System[Ref. 57]
92
Allen in 1957. [Ref. 58] The system stiffness effect during reentry is defined in the
quantity
MO (94)
where
ME- = the derivative of the dimensional aerodynamic moment with respect tothe angle of attack
1,. = reentry body's moment of inertia relative to the Z-axis
Based on these facts, Duzmak states:
The value of M." increases by several orders of magnitude upon the descentbecause of the increase in density and the indicated effect of the variation in thesystem's parameters is the main factor determining the damping of the oscillation... investigating the disturbed motion upon atmospheric entry permit one concludethat a descending entry body represents a significantly nonlinear mechanicalsystem. [Ref. 56:p. 7]
a. Equations of Perturbed Motion
The basic development of the equations of perturbed motion of a body
about its center of mass during reentry is prest.ited due to the uniqueness of Duzmak's
work [Ref. 56:pp. 12-28]. Equations (1) through (3) describe the system of equations
of motion for this problem.
The characteristic time intervals for the reentry body's motion relative
to its center of mass and the motion of its cen:er of mass are, respectively, defined as
(95)
93
Ar (96)V
where
f0 = characteristic rotational velocity with respect to the cente; of mass
Ar = characteristic variation in the radius vector of the center of mass
V = characteristic velocity of its center of mass
In the development of the equations of perturbed motion, the following
factors and assumptions were taken into account:
1. The interaction between the reentry body's motion with respect to its center dmass and the motion of the center of mass is neglected.
2. The hypothesis is satisfied in the upper atmosphere portion of the descenttrajectory until the angles of attack becomes less than one radian because of theatmosphere's stabilizing effectt.
3. The hypothesis is satisfied in the upper atmosphere portion of the descenttrajectory since the body's kinetic energy of the center of mass is many timeslarger than the body's kinetic energy due to its motion relative to the center fmass. Due to this fact, the aerodynamic moments begin Lo affect the motionabout the center of mass by stabilizing the body at significantly higher atiudesthan the aerodynamic forces acting on the center of mass.
4. The center of mass trajectory parameters are known as a function of time.
5. The flight trajectory is two-dimensional.
6. The average rotation of the velocity vector of the reentry body's center of masswhich is the curvature of the average trajectory is neglected. This effect iscausel by the gravitational force and the body's rotational velocity about itscenter of mass.
7. The hypothesis is clearly satisfied for the sections BC and DF as shown inFigure 26 [Ref. 56:p.16], because the trajectory is nearly linear due to flightvelocities of several kilometers per second. In the sections AB and FG, where
94
40
01--"'ZTrnosphere' s
.AWiN
Figure 26" Trajectory Of The Reentry Body[Ref. 561 95
the trajectory inclination angle, 9, varies significantly, the hypothesis is usuallysatisfied. The hypothesis is not satisfied in the exoatmospheric section CD.
8. The variation in the velocity vector orientation caused by the effec, theaerodynamic lift force is taken into account.
The reentry body coordinate system, shown in Figure 27 [Ref. 56:p. 23],
has both dynamic and aerodynamic symmetry and is used for the derivation of the three-
dimensional perturbed equations of motion:
1. The OXYZ right-handed coordinate system is fixed in both inertial space and
relative to the reentry body.
2. The body's center of mass is denoted by, 0.
3. The OXc axis is tie velocity vector of the center of mass.
4. The angle, a, between the OX and OXc axis is not the three-dimensional angieof attack or mutation angle.
5. The plane containing the OXc, OX and OY axis is the angle of attack plane.
6. The motion of the body in the angle of attack plane is defined by dx/dt and isdirected along the OZ axis.
7. The rotation of the angle of attack plane with respect to the velocity vector,OXc, is determined with the help of the precessional velocity, X, which isdirected along the OXc axis.
8. The body's rotation with respect to the angle of attack plane is determined usingthe intrinsic rotational velocity, 1,, directed along the OX axis.
9. The angle, -y, between the trajectory plane and the angle of attack plane isdefined by d-y/dt = X.
10. The angles a and -y have the following ranges: 0 < a < 19,0; 0 < -y < o.
96
armmTrajectory of theanlcenter oo $
\Xc
zTrace of theTTrace of the
trajectory'splane
Figure 27: Reentry Body Coordinate System[Ref. 56]
97
This coordinate system can also be used to describe two-dimensional motion about the
center of mass.
The angular velocity vector, w, of the reentry body is the resultant of
three rotations: da/dt; X; and g. As previously mentioned, the aerodynamic lift force
affects the body's velocity vector orientation. Lift acts in the angle of attack plane,
creating a motion of the body's velocity vector that is in the same plane with the change
of the angular velocity along the Z axis and is defined as
L (97)
where
L = force of lift from equation (12)
11 = mass of thc body
= tt, (small parameter)
The reentry body's angular velocity vector X, Y, Z components are
defined as
W = / + X cos a (98)
= -X sin a (99)
dx L (100)
The reentry body's angular momentum vector, 11, XYZ components are
defined as
98
H (101)
HY (102)
H• 1-,% (103)
where
reentry body's mnordents of inertia along the X, Y, Z directions
The main moment force, M, acting on te reentry body consists of a
restoring moment, M,(r, u), and a small damping moment that can be projected in the X,
Y, Z directions, is given by
M(r,(a) •- mqAL (104)
z qAL 2 (105)
M'y(r, a qAL 2 (106)M;•~r') =YY- V
M,7'Z(rc) M z qAL 2 (107)
V
where
M, , m.x , myy , mtz dimensionless aerodynamic (108)coefficients
q pVl/2
7 Et
99
By projecting dH/dt = M from equation (3) on the OXYZ coordinate
system and taking equations (101) through (107) into account, the reentry body's
equations of perturbed motion can be written in the form
SEM (109)
dw
u +W(11 1)
1ýdt =Z ,( .C0+E .Z MWz
where
W" = Xcosa
WTy =" CJy
Wz = WX
Equations (109) through (111) can be rewritten by substituting in equations (99) and
(100) for w, wy, and w,, and by substituting w, from above, into the following form
dr +cf (r, o)r = 0 (112)dt
dA + (2•coso -r). dot +, f~,) L r 0 (113)
Ssind dI io
d'a 2 since cosca +rX sinck +M(r,o) +fZ(ra)- dot = 0 (114)
dt10
100
where
r
M(7,c0) M - ',a) (116)
-M.x(r,cc) (117)
Ma) M y(T,a) L COS a (118)( ) Iz + mV sin-
Mf z(r,)) LU (119)fIrz) _g __.___
L c L (120)
Finally, the system of equations that describes the two-dimensional
motion of the reentry body's center of mass is given by
d - CL cOS'WqA g (121)S=• - COS 0 1 - VI-dt mV T
= - C -gr sinO0 (122)
101
d =I eV sinO (123)dt
dL . Ro V cos0 (124)
where
gT = gravitational acceleration of the Earth
0 = angle between the tangent to the trajectory and the local horizontal
H = altitude of the flight
R = Ro+ H
L = range of flight measured with respect to the Earth's surface
Tie investigation of equations (112) through (114) in the case of known
solutions for equations (121) through (124) is the principal content of Duzmak's work.
Specifically, the following major areas were investigated:
1. The distinctive features of the two-dimensional motion that explains theinteraction of the nonlinear effect with the influence of variability of parameters.
2. The distinctive features of the motion that results from the transition through thetransonic velocity range.
3. The relationship of the angular momentum components with the parameters cfmotion in a increasingly denser atmosphere. This analysis clarifies the effectof the shape of the instantaneous characteristics, stability margin and damping.
4. The effect of the reentry body's motion parameters above the atmosphere on itsmotion in the denser layers of the atmosphere.
5. The case of sinusoidal dependence of the longitudinal moment on the angle ofattack for three-dimensional motion.
102
6. The bcoa-(,s motion relative to its center of mass for a small angle o,' attack
9m
reentry (shallow angle reentry from a decaying orbit).
b. effect of Motion Retve to the Center of Mass on the Motion of the
Center of Mass
The coupled effect of the perturbed motion about the center of mass with
the trajectory of the ceater of mass occurs because of the dependence of the aerodyn 1mic
coefficients on the angle of attack. Duzmak neglects this interaction in the development
of the asymptotic solutions. This assumption is based on the fact that the atmosphere
will start to influence the motion relative to the center of mass at higher altitudes than
the atmosphere will start to effect the motion of the center of mass. Generally, for
perturbed motion abo,!cut. t ,.nte.r Of mss M, a .. .c-imolccuar regime, t1hic cf"f%,c f" th•
lift force is small, since it continually acts in different directions. In the denser layer of
the atmosphere, lift can also provide some additional damping of the angle of attack
oscillation. [Ref. 56:p. 29]
The main effect on the trajectory is exerted by adr,, equation (6). For
perturbed motion at high altitudes where atmospheric density is low and C,(a) varies
significantly, a, has little effect on the trajectory. In many cases at lower altitudes,
perturbed motion has a weak effect on the trajectory due to the fact that a reentry body
is:
... able up to this instance to stabilize itself under the action of aerodynamicmoments so that the variation in C. in the case of perturbed motion becomesinsignificant. [Ref. 56:p. 29]
103
For the case of hovering or similar motion where the oscillations do not
dampen, the oscillations of the angle of attack can occur with significant amplitude.
Under these conditions, i, is essential to take into account the angle of attack oscillation
effect on the trajectory of the cente, of mass. [Ref. 56:p. 317)
The determination of the range component of the scattering landing points isimportant. This scattering is caused by the oscillations, the angle of attack and thevariation of the coefficient of drag associated with the oscillations. This quantitycannot in general be determined without taking into account the interaction ofmotion relative to the center of mass and the motion of the body center of mass.[Ref. 56 :pp. 317-3181
In the derivation of a simple approximate method to solve this problem,
the descent trajectory was divided into two sections: H > Ib (trajectory in the
atmophere's tenuous layers) and H _< Hb (trajectory where the interaction between the
motion with respect to the center of mass and the center of mass trajectory is taken into
account). 11b is the limiting altitude that is calculated to be 70-80 km.. In the region,
II !:- Hb, the refined asymptotic nmethod breaks down. An approximate solution is
derived by formulating averaged equations for the asymptotic solutions that describe the
change of the slow varying components of :
1. The maximum and minimum values of the angle of attack during ev;hoscillation
2. The instantaneous oscillation period
3. The averaged components of the center of mass trajectory
The derivation of this method is mathematically rigorous. [Ref. 56:pp. 319-336]
104
The direct solution to this problem can be obtained by numerically
integrating the complete equations of motion given by [Ref. 5 6 :p. 451
a, 1 + ME (xix2,... a,) + Efz(xIIx2, .... ,A) do 0 (125)
dxt- +FXx ,X2,..... c 0dt (126)
where
= (G-r cosa)(r-G cosca) +M(rU) (127)sin•a
X, = f (r,a) G + ([f(T-a) -f (r,a)] cos ot -L(r,ce) sin a) r (128)
X2 = f(7r,a) r (129)
3= M +g, sinO (130)
=g, Cos0 i-- (131)
X = -V sin 0 (132)
X6 = -R. V cos0 (133)
105
G - Q cosa(-QY sina = X sin 2a +r coso = constant (134)
k
However, in 1970, this numerical integration required an excessive amount of machine
time or computer time, since the solution contains high frequency components along the
trajectory.
c. Follow-on Investigations of Uncontrolled Reentry Body Motion
References [591 and [60] are extensions of Duzmak's investigation. Each
paper examines a certain aspect of the uncontrolled three-dimersional motion of a reentry
body relative to its center of mass.
Reference [59] investigates the three-dimensional uncontrolled motion of
a spherical body relative to its center of mass with an arbitrary angle of attack. The
mean differential equations a~e derived for rolling motion. These equations are ,ed
in "mplicit form for any angle of attack by u.ing the Van der Pol method of integration.
Reference [60] investigates the three-dimensional uncontrolled motion
relative to the centei of mass of a reentry body with a small geometric and dynamic
asymmetry. An approximate analytical solution for the equations for unperturbed three-
dimensional and the averaged equations for perturbed three-dimensional motion are
derived. by imposing no limits on the initial angular velocity and the three-dimensional
angle of attack or nutation angle, the averaged equations of motion computational time
is reduced by a factor of approximately three as compared to the numerical integration
of the complete equations of motion.
106
B. SIX-DEGREE-OF-FREEDOM SIMULATIONS
The foundation for six-degree-of-freedom motion has been discussed in the previous
section. Several other investigators have examined the effect of motion relative to the
center of mass on the trajectory of the center of mass [Refs. 61-64]. The equations of
orbital decay, (121) through (124), when coup!ed with the equations of attitude motion,
(112) through (114), will completely describe the motion of a vehicle during the final
stages of life. Equations (125) and (126) are the coupled equations of six-degree-of-
freedom motion written as a function of angle of attack.
Coupling these equations means: the solution of one set of equations determines the
magnitude and form of the forces or moments in the other set [Ref. 6 2 :p. 13]. For
example, if the area changes due to either a loss in mass or a change in attitude, then the
ballistic coefficient (W/CDA) and the coefficient of drag will be affected.
References [61], [62] and [63] investigate Skylab's attitude and trajectory motion.
Specifically, references [62] and [63] derive a variation of the coupled equations
presented in the previous section. The authors performed a six-degree-of-freedom
trajectory simulation using the coupled equations of motion. Figure 28 shows the flow
chart used in their simulation [Ref. 62:p. 141. Numerical integration of the six
differential equations yielded altitude and orbital elements as a function of time.
Computer run results for these simulations were extremely long and therefore the
decision was made to artificially increase the magnitude of the aerodynamic damping
Subsequently, the results do not simulate the actual dynamical behavior, but they show
the "possible" types of dynamical behavior for Skylab. [Ref. 62:p. 15]
107
Initial
Conditions
Atmospheric
Density
(Jacch~ia, 1970)
F-Re-rodynami1L and ý inLGravity Gradient171
,Euler'. Equ~aciono(with damping)
rNumer1.calIntegration
Altitude
Orbital Position ft)_
Figure 28: Six-Degrec-Of-Freedorn Simuilation Flowchart[Ref. 62]
108
Reference [64] is a reentry analysis of a proposed radiois3topic thermoelectric
generator (RTG) that was connected to a generic user satellite. A 3-degree-of-freedom
reentry trajectory simulation was conducted that utilized aerodynamic, material property
and heating characteristics.
Newtonian and free molecular drag coefficients were calculated using the Mark IV
Supersonic-Hypersonic Arbitrary-Body program for the generic satellite and each of its
subelements. Figure 29 shows the three-dimensional shapes of the generic spacecraft and
one subelement (reactor with radiator) u ed by the program [Ref. 6 4 :p. 6]. Aerodynamic
blockage was neglected and a zero angle of attack was assumed in the calculations. The
heat transfer methodology used in the simulation to predict the satellite's breakup during
reentry, implemented a calibrated heat transfer model to closer simulate actual conditions
[Ref. 64:pp. 7-i2]. This wiil be discussed further in thie section on breakup later in the
chapter. The trajectory simulation was designed to change the mass properties and
aerodynamic coefficients as the shape changed due to an "assumed predetermined
breakup" scena-io. This assumption was based on the basic geometric components or
subelements of the satellite and their probable separation sequence during breakup. For
example, the boom separated from the main part of the spacecraft and the reactor failed
first, this was followed by the heat radiator cone, and then the other components in
sequence. [Ref. 64:p.14]
Trajectory simulations were run to assess the breakup altitudes, due to reentry
heating, and minimum footprint lengths, due to fuel pin release at various a, itudes,
which were independent of the heating effects.
109
GENERIC SPACECRAPP
REACTOR WITH RADIATOR
LL
Figure 29: Simulated Generic Spacecraft And Reactor Sub-Element[Ref. 64]
110
C. RAREFIED GAS DYNAMICS
Numerous authors have investigated reentry vehicle attitude, heating rates and
aerodynamic coefficients in relation to the atmospheric flow regimes [Refs. 22, 56, 65-
75]. As statea in chapter II, the changes in flow regime and corresponding changes in
critical parameters of the reentry trajectory or heating equations, are poorly understood
in the transition from free-molecular Lo hypersonic continuum flow. For engineering
applications the quantities of lift, drag and heat transfer are usually estimated by a
"judicious faring" between regimes [Ref. 22:p. 203].
One solution to this problem is presented by Koppenwallner and Johannsmeier,
reference [76]. This solution is a technique of "bridging" between the free molecular and
continuum flow, based on Newtonian and free molecular theory. This technique allows
the prediction of lift and drag during the hypersonic entry phase. [Ref. 76:p. 461]
Three hypersonic flow reginies, with approximate boundaries, are described as
follows: [Ref. 76:p. 461]
1. Free molecular flow Kn > 5
2. Rarefied transitional flow 5 > Kn > 0.001
3. Hypersonic continuum flow Kn < 0.001
These boundaries are actually dependent upon reentry vehicle shape and on the
aerodynamic property considered. Figure 30 shows these flow regimes in an altitude
velocity graph [Ref. 76:p. 461].
Ill
""-
)<n *5
1000 4 o 12 20I-
C-
reference length hIsm ZC
-"I f
ePVV -e.
0 1 2 3 4. 5 6 8 km /s 10
Figure 30: Flow Regimes - Altitude vs Velocity[Rcf. 76]
112
The analysis of this technique is limited to flow conditions where the hypersonic
Mach independence principle is applicable; therefore, blunt shaped reentry vehicles will
have a lower limit of Mach 5. Figure 31 shows the typical aerodynamic data variation,
for a blunt body, through the three flow regimes of interest [Ref. 76:p. 461].
The typical drag coefficient behaves such that in free molecular flow, the drag
coefficient is independent of Knudsen number and the value of CD = 2. The drag
coefficient decreases throughout the transitional flow regime and again reaches a constant
value in the continuum flow regime. [Ref. 76:p. 4611
The typical lift coefficient behaves such that in the free molecular flow, the lift is
very small or negligible. The lift coefficient increases throughout the transitional flow
regime until it reaches a hypersonic continuum flow value. [Ref. 76:p. 461]
The heat transfe' Stanton number (St) is very close to a value of one in the free
molecular flow regime. In the transitional flow, the heat transfer efficiency decreases
and the Stanton number decreases. In the continuum flow, the Stanton number decreases
continuously as a function of Reynolds number in the manner St- 1/VRe. [Ref. 76:p.
461]
Table III describes the bridging dependencies as modeled above. The derived
transitional functions D2 and L2 are functions of angle of attack, vehicle shape and
Knudsen number. These transitional functions must bridge the free molecular and
continuum flow regimes and must degenerate, in the two limits, to the free and
continuum flow values as shown in Table IV. [Ref. 76]
113
lconl"
f ree 1rarefied cotnuummolecular transition f low
2ost co z fKn)p.d
Cont con s-.1 _______ _______
Co |pressureNdrag only)
KnzOS 00020 .... . - -_ I -_ -__ _ _ .I . - - 1 --- t
10 1 01 0.01 0.001 Kn0
free rorefied continuummolecular transition f iow
I ST:¶ST: const,.0.1 [ ~const. /~"
ST ifiactual behoviour
0-001- - pproximation
10 1 0.1 0.01 0.001 Kn,,--
Figure 31: Aerodynamic Coefficients In The Flow Regimes
[Ref. 76]
114
Table III: BRIDGING DEPENDENCIES[Ref. 76]
FLOW REGIME DRAG LIFT
Free molecular CD = D, (a, shape) CL = 0
Transitional CD = D2 (a,shape, Kn) CL = L2 (u, shape,Kn)
Hypersonic CD = D3 (a, shape) CL = 1.3 (a, shape)continuum
Table IV- TRANSITIONAL BRIDGING FUNCTIONS[Ref. 76]
DEGENERATE TO TRANSITIONAL DEGENERATE TOFREE MOLECULAR BRIDGING FUNCTION HYPERSONIC CONT.
D = (a, shape) D2 = (a, shape, Kn) D3 = (a, shape)
L 0 L2 = (a, shape, Kn) = (a, shape)
Local flow independence exists in free molecular as well as hypersonic Newtonian
flow. This means that unless "shadowing" exists, surface and shape elements will not
influence each other. The shape elements of this technique are cones, spherical caps and
cylindrical shells. These shape elements allow the composition of capsules, blunted
cones and cone-cylinders. Figure 32 shows the shape elements and several composed
bodies [Ref. 76:p. 462].
115
basic elements composed shapes
... ~P~rI~t . } ciap-Cylinder
Cj d cone, cone -cylinider
cylinder d capsule
Figure 32: Shape Elements And Composed Bodies[Ref. 76]
116
1. Newtonian Aerodynamics for Hypersonic Continuum Flow
For the case of blunt bodies, it can be assumed that the main contribution to
drag is pressure; therefore, the influence of viscous effects on the aerodynamic
coefficients are neglected. [Ref. 7 6:p. 462] Using the Newtonian pressure law, the
coefficients of drag and lift may be described in differential equations. The basic
Newtonian relationships are as shown in Table V. [Ref. 76:p. 462]
Table V: NEWTONIAN I OCAL PRESSURE LAW[Ref. 76]
Wetted surface 0 < 90g p / q_= k. co.
Newtonian wake I0e > 900 p / q_= CD.
where
p = pressure
q., = dynamic przssure in free stream
k• = Newton factor
0 = flow inclination against surface normal
K = ratio of specific heats
CD0 = coefficient of drag as a function of angle of attack
117
and
kN = 2 (simple Newton)
kN = (x + 3 )/(K + 1) (modified Newton)
In the Pike formulation, the Newtonian differential equation is given by [Ref. 71:p. 462]
C )cot()+16k A(c) (135)
CDQ(a) +CýOCO)+ l2 CDc) kN AR
3CL(cr) = CD. (136)
where
Ap = Flow projected area of body
AR = Reference area (a-D2/4)
In order to determine the drag coefficient and solve the differential equations,
the following approach is taken: C, (a=0°, shape). This implies that the zero angle
of attack drag is proportional to the Newton factor and to a shape dependent factor, Cs
Ref. 76:p. 462]
C (a=-0 ) kN Cs (137)
Upon integration of the Newtonian pressure distribution, the following values
for C. are determined for various bodies in Table VI. [Ref. 76:p. 462]
H18
Table VI: SHAPE FACTOR VALUES[Ref. 7611
Disk Cs= 1.0
Sphere Cs= 0.5
Cylinder Cs= 0.67 (cross flow)
Cylinc-rIcal shell CS= 0 (parallel flow)
Sharp cone Cs= sin20-
Spherical cap Cs= 1 - 1/8 • (d/rN) 2
Fizure 33 shows the shape factor as a function of the ratio (ir~,,), diameter
over nose radius [Ref. 76:p. 466].
When the shape factor or drag at zero angle of attack are known, the
aerodynaraiic coefficients of the class are completely fixed. The following universal
solutions for drag and hf' obtained from the Newtonian differential equations [Ref.
76:p. 462]
119
1.0
40.9T46II r"d
08~
CS C I d' I'z
0,7,
0 05 1 1.5 2d ---- _
Figure 33: Shape Factor As A Function Of (d/r,)[Ref. 76]
120
C =-- (2C. - (5Cs - 3) sin2(0)) \ (138)
CL= KtL ( 2 (1 - 2C5) - (5Cs - 3) sin2(a))sina (139)
These solutions are valid for a : o, , since at oe,. the wette area decreases due to
Newtonian shadowing. Figure 34 shows the universal Newtonian iift and drag functions
for various angles of attack [Ref. 76:p. 466].
The shape factor at zero angle of attack serves as the critical parameter in this
method. The solution is valid only under the condition that the wetted surface area
remains constant for any angle of attack. [Ref. 76:p. 462]
As the geometric body bluntness increases, Cs increases and Newtonian
shadowing is shifted to higher angles of attack. Table VII shows that depending uDnon
the body shape, the lift slop:; at zero angle of attack may be either positive or negative.
[Ref. 76:p. 462]
121
0.80
co/k. Q2
CLikN 0.20
-0.
- 0.24.
0* 100 20" 30' 40' 500 60" 70* 800 9Q4a
Figure 34: Newtonian Lift And Drag Functions[Ref. 76]
122
Table VII: BODY SHAPE AND LIFT SLOPE[Ref. 761
ri
Cs Body shape Lift slope @ a=O
< 0.5 slender positive
> 0.5 blunt negative _ _
This demonstrates that hbunt reentry bodies wi~l experience a negative lift for a positive
defined angle of attack. [Ref. 76:p. 462]
2. Free Molecular Flow Model
The technique of reference [76] also develops analytical formulas for free
molecular flow. The general considerations include: [Ref. 76:p. 463]
1. Accommodation coefficient, a-
2. Finite molecular speed ratio, S
3. Wall temperature, T,
These factors alone are insufficient thus the following simplifying assumptions are made:
[Ref. 76:p. 463]
1. Body shape Hypersonic blunt, Ma (d/l) - 2
2. Acrodynamic cold wall Tw/ T., < < 1
3. Diffuse molecular reflection a = 1
123
From these constraints the local surface pressure and shear stress may be
defined as shown in Table VIII. The aerodynamic force coefficients are also derived as
shown in Table IX.
Table VIII: LOCAL SURFACE PRESSURE AND SHEAR STRESS[Ref. 76]
Wetted surface 101 < 900 p/q.-- 2 cos 20
Unwetted surface 101< 90 q.= sinO*cose
Table IX: AERODYNAMIC FORCE COEFFICIENTS[Ref. 761
Drag coefficient Cc= 2.Ap(a) / Ap(O) = cos a
Lift coefficient CL= 0
These assumptions imply that no lift is produced and drag is proportional to the flow
projected area, Ap(a), in the free molecular flow regime. [Ref. 7 6 :p. 463]
In the more general free molecular flow case, where Tw/T. , oa. and a, are
considered, an axisyrnetric blunt body formulation for the force coefficients may be given
by
124
CD = 2acosof + a. _- C + + c,,(140)
"" T (141)
(2 -a,)•- -(Cs, + )sina --(5 Cs2- 3)sin3ci]4=
where
Tw. = temperature, free stream
Cs, = shape coefficient, in front of normal shock wave
Cs2 = shape coefficient, behind normal shock wave
The physical significance of several of the terms in equations (140) and (141)
are: [Ref. 76:p. 463]
1. The first term in CD gives, for diffuse reflection, a= 1, the contribution of theincident flux to the aerodynamic coefficients.
2. The Te/Tu. term states the influeoce of the reentry vehicle surface walltemperature on drag and lift.
3. The (2-u0 -u,) term vanishes for diffuse molecular reflection.
In simple free molecular flow the equations will degenerate appropriately as
= 1, a,= 1 and Tw/T,. =0. In the Newtonian formulation the equations will degenerate
as ha= 1, a,=0 and TW/T,ý =0. In this last case C., vanishes and CQs is the Newtonian
shape coefficient, C,,. [Ref. 76:p. 463]
125
3. Bridging Free Molecular and Continuum Flow
Several methods exist which attempt to bridge the gap between the free
molecular and continuum flow regimes. [Refs. 77-79]. In the former USSR and DLR,
local bridging is accomplished with a finite surface element method, [Refs. 77-78]. In
the U.S., reference [79], bridging has been accomplished through an integral coefficients
method. [Ref. 7 6 :p. 463]
The method of bridging by shape element description is developed in
reference [76] and is presented in the following section. The methods of references [78]
and [79] may be used to derive analytical formulas for trajectory calculation. However,
the basic bridging relations must be derived through experimentation. [Ref. 76:p. 463]
a. Shape Element Bridging Method.
Experimental data, presented in Figures 35 and 36 show the drag
coefficient changes of a sphere and disk respectively. The data covers the the entire
transitional flow regime [Ref. 76].
For a sphere, the Reynolds number, behind the normal shock (Re2), can
be related to the Knudsen number, in the free stream (Kn,), as follows [Ref. 76:p. 463]
Re2 =1.26 f 1 (142)
From this experimental data, an approximate formula for Ci,, as a
functior of Reynolds number and shape, can be derived by using the "reduced"
aerodynamic ccefficients. [Ref. 7 6 :p. 463]
126
ts-U, O I i 1
ii Lion- A 669 ~ fre let NvvL)
3 -p 1JO0w39 .jm~ao
,O o _ l•O e +3 .. W , ca - -ozi
130 1• peulum metn<d.
001 01 10 10 100 1000Re 1 - __ - ::,
Figure 35: Sphere Drag Coefficient In Rarefied Flow[Ref. 76]
127
ACevdw ~ 3O~C 9?I1c~n:91 o
Fiur 36 isk DragCoefcetIaeidFo[Ref. 76]
1 OLEI I ota WA CM 12C8
CD(Re) - CX (143)
DCDFM -CDC
7 L/D(a = const., Re2) (144)LID (a = const. , Continuum)
where
C~c -= drag coefficient, continuum
CDF = drag coefficient, free molecular
CD = reduced drag coefficient
L/D = reduced lift-drag ratio
Analysis of the reduced coefficients at zero angle of attack shows the
following: [Ref. 7 6:p. 463]
1. Slope of the reduced draag coefficient is shape_ i..dependent.
2. Continuum and free molecular flow boundaries are shape dependent.
3. Continuum ana free molecular drag coefficients are shape dependent.
Therefore the bridging function is derived from the reduced coefficients as follows [Ref.
7 6 :p. 464]
CD 01 (cDF CDln Re2 +CCD=5.205 Re 2 C
And from this, the drag coefficient may be extracted in one of two forms
129
CD I K + CC (146)
CD= CD(Re2)- CD 1 In Re 2C (147)CDom-Cc 2.26 Re2
The shape dependent boundaies are defined as:
1. Re2c > Re 2 > Re-mm
2. Knc < Kn > Kn~m
and the experimentally derived values are summarized in Table X below. [Ref. 76:p.
464]
Table X: SHAPE DEPENDENT BOUNDARIES[Ref. 76]
Body Continuum Flow Free Molecular FlowShape d/RN Re 2c Knc Re2FM KnFM
Disk 0 28.5 0.117 0.115 21.5
Sphere 2 89.3 0.037 0.492 6.8
In summary, the technique of reference [76] allows the following
conclusions to be drawn: [Ref. 76 :p. 464]
1. Newtonian theory based on Pike's method for the hypersonic continuum flowis useful, as it shows the shape dependent and shape independent aerodynamiccoefficients.
130
-LI
2. The bridging function is based on experimentally derived results for thetransition flow.
3. Lift and drag bridging do not follow the same laws.
b. Local Bridging Method
Reference [78] develops a technique of "local" bridging. The differences
between the shape element method, referred to as a "global" method, and local bridging
are: [Ref. 7 8 :p. 469]
1. Global bridging performs across the spectrum of aerodynamic coefficients forthe complete body.
2. Global bridging is usually tailored to a specific class of shapes.
3. Global bridging techniques require new experimentally derived "fittingCornsztnt.ts" for each new zhvape of interest.
4. Local bridging is a method of bridging the local pressure and shear stresscoefficients. The local distribution is then integrated over the body surfacewhich yields the global force and moment coefficients.
The common principle of all bridging techniques is the manner in which they model the
transition function. The known free molecular and continuum flow limiting values for
a specific aerodynamic coefficient (lift, drag or heat transfer) are weighted and applied
to the bridging function. The general case is as follows
C(X) = CFMJAX) CcC! -fX)) (148)
131
where
C aerodynamic coefficient considered
X = rarefaction-dependent flow parameter
The potential improvements of local bridging over global bridging are:
[Ref. 78:p. 470]
1. Ease of adaptation to different shapes without changing internal constants.
2. More reliable calculation of moments. These are very sensitive to rarefactionbecause of the varying contributions of pressure and shear in the different flowregimes.
3. Simplified method for determining reference quantities for local coefficients,such as reference length based on a local coordinate.
4. Ability to account for non-uniform flow field conditicns.
The conclusion of reference [78], after examining several different
bridging methods and various shapes, is that the current state of experience in applying
the local bridging methods allows no definitive answer as to whether or not they can
serve as an effective analysis tool in transition flow analysis.
4. Gas-Surface/Gas-Gas Interactions
Gas-surface interactions are significant in the understanding of reentry
dynamics and aerothermodynamics. The primary cause of concern is that at orbital
altitudes, the highly rarefied flowfield is dominated by gas-surface interactions that occur
at average velocities corresponding to that of the orbiting vehicle. [Ref. 80:p. 1] Under
these reentry conditions, gas-gas interactions become important and gas molecules that
i_32
reach the vehicle surface tend to have lost some of their initial translational energy. This
loss of translational molecular energy is due io conversion into other forms such as heat,
internal energy, etc. Figure 31 shows molecular velocity distributions in rarefied and
transitional flow about a reentering sphere [Ref. 80]. The collision mechanisms by
which the translational energy is converted include:
1. Chemical reactions
2. Ionization
3. Dissociation
Since the nature of the gas-surface interaction is known to be dependent upon the velocity
and energy of the incident molecules, it becomes necessary to know the state of the gas
molecules reaching the surface. [Ref. 80:p. 1]
Wilmoth, et al, [80] uses the technique of Direct Simulation Monte Carlo
(DSMC) as developed by Bird, [Ref. 81] where the molecular velocity and energy
distributions of the gas molecules are a direct result of the s~mulation process. Bird uses
a simple engineering model of the gas-surface interactions, which accounts for diffuse
and specular reflection along with other phenomena. This model can accommodate
processes such as catalytic reactions and molecular recombination. A limitation of this
method is that paramentric studies must often be performed in order to place bounds on
the predicted quantifies of interest. [Ref. 80:p. 1] This also implies that the analyst must
judiciously apply the model based on experience and a limited base of experimental data.
133
SHOCK LAYER
Cq~~Ins
P~P) (AX. .\P) Reflected... ~.,%.,Velocity
Freestream 1J istiutoVelocity V
Distribution 1
Figure 37: Molecular Velocity Distribution[Ref. 80]
134
The lack of experimental data is the reason why more detailed gas-surface
interaction models have not been developed. This results mainly for two reasons: [Ref.
80:p. 1]
1. It is very diffictlt to simulate gas-surface interactions at orbital or entryvelocities in a laboratory.
2. It is difficult to characterize the surfaces used in laboratory experiments withsufficient generality that the results may have application in an engineeringcontext.
Because of the tendency of gas molecules to decelerate after gas-gas collisions, which
occur before reaching the vehicle surface, it is important to quantify the actual velocities
and energies encountered in such cases.
R~efe-rence [801 studieS the "tvnircl" gas-surfac. " o for transitiona
flow at entry velocities. The reentry vehicle was a 1.6 m diameter sphere at a free
stream velocity of 7.5 km/sec over an altitude range of 90 to 130 km. [Ref. 80:p. 2]
The study was conducted using the DSMC method of 3ird.
The flow conditions of the DSMC simulation are given in Table XI [Ref.
80:p. 6]. The atmosphere is modeled by Jacchia, 1977, with an exospheric temperature
of 1200K. The surface temperature of the sphere was assumed constant at 350K. The
gas-surface interaction was assumed to be diffuse with full thermal accommodation, and
the surface was non-catalytic. Five atmospheric molecular species were modeled 02, N 2,
0, N and NO, with 23 reaction possibilities. [Ref. 8 0:p. 31
135
Table XI: FREESTREAM CONDITIONS[Ref. 801
_ -, _ , I ll_ - ro=
Afitud p. v. T. Mole Mole Mole M A.km kg/rnm kmn/s K Fraction Fraction Fraction g/mol m
0, N, 0
90 3 43x104 7.5 188 0.209 0.788 0.004 28.80 C.017
100 5.6ex107 7.5 194 0.177 0.784 0.040 28.24 0.100
110 9.67x104 7.5 247 0.123 0.770 0.108 27.22 0.599
120 2 27x10' 7.5 388 0085 0.733 0183 26.14 2.681
130 8.23x10' 7.5 500 0.071 0.691 0 236 25.43 7.724 J
Wilmoth's results show the following: [Ref. 80:pp. 3-5]
1. At 130 kin, a density rise of nearly 22 times the freestream value occurs overa distance of - 3 m. However, based on analysis of the data it is determinedthat very few collisions are occurring.
2. At 90 kin, the density increases to well over 100 times the freestream value overa distance of - 0. 1 nn. In this altitude range collisions become very significantand there is considerable chemical activity.
3. There are significant variations in the velocity and energy of molecules reachingthe surface over an altitude range of 130 to 90 km. Figure 38 shows theaverage translational energy pe- particle striking the reentry vehicle surface[Ref. 80].
D. SURFACE ROUGHNESS EFFECTS
Another important consideration in the reentry phase is the "boundary-layer
transition." It is this phenomenon which is responsible for heat transfer to/from the
reentry body due to atmospheric contact with the body. In the continuum flow regime
viscous effects are essentially restricted to a small layer called the boundary layer. It is
136
Altitude, km
101 go 100 110 120 130
A
Limits VSW.C
*-c--17o- -- All Species
101........ -At~fR 1 A..11
Knudsen Number
Figure 38: Average Translational Energy[Ref. 80]
137
within this boundary layer that the details of "fluid" motion deteimine the levels of skin
friction and heat transfer from the flow. [Ref. 22:p. 211]
A parameter commonly used for characterizing the boundary layer is the Reynolds
number. [Refs. 82-83] The Reynolds number is defined as the ratio of inertial forces to
the viscous forces, as given by [Ref. 2 2 :p. 212]
inertia forces - A (momentum) / (unit time) _ Re (149)viscous forces (shear stress )x (unit area)
In order for a fluid (the atmosphere is modeled as a fluid in the continuum flow)
to support a shear stress there must be relative motion between adjacent layers. This
implies that there is a velocity differential within the flow layers. The following terms
are defined: [Ref. 22:p. 213]
T = shear stress = p.(cIdly)
A = dynamic viscosity
y = coordinate normal to direction of motion
The definition of Reynolds number may now be given as [Ref. 22:p. 2131
Re d(mV)/di [pL3V/ (L/V)] VL VL (150)
4(dV/dy)A p(V/L)v
where
V kinematic viscosity = (pyI)
Now the Reynolds number may be used to characterize the boundary layer flow
conditions. When the viscous forces are large enough to damp out the oscillations caused
by the dynamic forces, the flow is laminar and the Reynolds number is small.
138
Conversely, the flow is turbulent when the dynamic forces overcome the viscous forces.
Also, the Reynolds number is large for turbulent flow. [Ref. 22:p. 212]
The velocity profiles within the laminar and turbulent boundary layers show some
significant differences. The magnitude of velocity in the turbulent boundary layer is
notably greater, especially near the reentry body's surface. This implies that the
turbulent boundary layer yields much more energy near the surface than does a
corresponding laminar flow. The velocity profile determines the skin friction on the
surface and it can be expected that greater heat transfer will occur under conditions of
a turbulent boundary layer [Ref. 22 :p. 213]. Figure 39 shows the boundary layer types.
Figure 40 shows boundary layer velocity-distance profiles. Figure 41 shows the altitude,
air speed and dynamic pressure [Ref. 22:pp. 213-214].
A Re-nolds stress turbulent boundary layer model which srpcifirally .accounts for
surface roughness effects is described by reference [84]. In this study, surface roughness
is represented by distributed "sources" and "sinks" in the various governing equations.
The most significant term is a sink term in the mean momentum equation, which
represents "form drag" on the roughness elements. [Ref. 84:p. 2]
A fundamental assumption of this model is that the flow around the individual
roughness elements (only distributed roughness is considered) is attached to the elements.
[Ref. 84:p. 3] The roughness elements provide a distributed sink, due to drag, for the
momentum equation and distributed sources for mean turbulent kinetic energy and
dissipati )n. This model also assumes that the roughness elements occupy no volume;
therefore, this assumption becomes more severe as the roughness density increases. In
139
TUR BULENT
TRANSITION
LAMI1NARn
SEPARATIONLAIRSUBLAVER
Figure 39: Boundary Layer Types[Ref. 22]
140
41 0w
cc
LAMINAR
0 .51.
[Ref 22
141m
SOL X S1V3SYd 'Unfs3Ud 31WVNAG
0
'U
C-4
ZJ3SIVJ)4 033dS UIV
L L
Figure 41: Altitude-Air Speed-Dynamic Pressure Relationship[Ref. 22]
142
order to compensate for this, the model has been extended to account for the blockage
effect of the roughness elements. [Ref. 84:p. 4]
The mean momentum equation is given as [Ref. 84 :pp. 4-5]
.fly) pU-+pV U -fly ) [LP + a (151)
dX 7 - ax TY 1 (15D1)a -( 7- -)Ip U 2C D 1 -TY2 T2 411j
The variables U, V, u/ and v/ are the reduced variables, under the boundary layer
approximation, as given in reference [85]. These variables and this equation will not be
derived here for the sake of brevity.
The major advantages of this model are: [Ref. 84:p. 5]
1. Solutions are obtained for both velocity and thermal variables.
2. Heat transfer is obtained directly, without invoking Reynolds analogy.
3. Finite difference solutions are obtained using the boundary conditions thatfluctuating quantities are zero at the solid wall and in the free stream.
Reference [84] concludes with a comparison of a smooth wall turbulent boundary
layer model and the developed rough wall model. The rough wall model is determined
to show that roughness spacing is more critical than roughness height, under the
conditions tested; however, the limited skin friction data obtained in the study cannot be
interpreted unambiguously. [Ref. 8 4 :p. 38]
143
E. REENr'RY HEATING EQUATIONS
In Chapter II, the fundamentals of reentry heating were discussed as presented by
the authors of references [20], [40] and [43]. Aerodynamic heating, as it applies to the
reentry of space vehicles, takes its roots in the work of Allen and Eggers, reference [65].
Reentry heating becomes an important consideration of the overall reentry process for
the following reasons: [Ref. 42:p. 139]
1. Structural performance of the reentry vehicle is dependent upon the dynamicpressures encountered during the reentry, which is a function of the reentrytrajectory.
2. Structural strength of the vehicle is a function of the stresses induced bytemperature gradients within the component materials.
3. Temperature gradients are proportional to the time rate of heat input andmaximum time rate of heat input.
Therefore, the three critical parameters cf the reentry trajectory are the total heat input
along the trajectory, the maximum rate of aerodynamic heating and the maximum
dynamic pressure. [Ref. 42:p. 139]
The mechanism of heat flow into the reentry vehicle during atmospheric entry was
first described in reference [65]. Since then, numerous combinations of reentry speed
regimes and aerodynamic shapes have led to the publication of numerous technical
reports. However, the basic aspects of aerodynamic heating during reentry are still the
same. The numerical factors for different heat transfer formulas and their ranges of
validity in terms of the regime of speed are the only variation among the numerous
authors. [Ref. 4 2 :p. 139]
144
The basic equations of reentry heating, equations (18), (19) and (20), are developed
with their simplifying assumptions as presented in Chapter II, pp. 46-48. These
assumptions yield the following limitations: [Ref. 42:pp. 141-142]
1. Neglecting radiative heat transfer from the vehicle or to the vehicle from thehigh-temperature air between the shock wave and the vehicle surface, is basedon the fact that the maximum allowable surface temperature is about the samefor a variety of reentry vehicle shapes. Thus, outward radiation from thesurface will be about the same. Neglecting the radiative heat transfer from thedisturbed air is a qualitative simplification and therefore negates the applicationof the equations to very blunt and heavy shapes at reentry speeds of 3 km/secor greater.
2. Neglecting the real-gas effects in the flow, most importantly dissociation, onconvective heat transfer is a good approximation for reentry speeds of up to 3km/sec. Nevertheless, this assumption is conservative and results in highercalculated heating rates than actual rates.
3. Neglecting the shock-wave boundary-layer interactions implies that the laminarskin-friction coefficient, on a flat plate at zero incidence, is being held constant.T his assuniputn is nut vaiid at reentry speeds over 6 kmisec.
4. Assuming a Reynolds analogy and holding the Prandtl number constant alsorestricts the validity of the equations to reentry speeds of less than 3 km/see.
For the cask- of low earth orbits and naturally decaying satellites, these assumptions
create severe restrictions. The circular orbital velocity may be calculated from [Ref.
17 :p. 38]
_ (152)C
145
where
r = radius from center of Earth
V, = circular velocity
Using this equation, it is possible to approximate the circular orbital radius of a
satellite traveling at an altitude of near 120 km or radius of 6499 km. This is the altitude
of concern for the focus of this thesis and it can be shown that the orbital velocity of
interest is approximately 7.8 km/sec. The orbital radius of a satellite travelling at speeds
of 3 km/sec and 6 km/sec are 44,289 km and 11,072 km respectively. This equates to
circular orbital altitudes of 37,910 km and 4,694 km.
The assumptions of reference [651 which are maintained in references [14], [40]
and [43] should be removed for an accurate quantitative analysis of the aerodynamic
heating durin.g the reentry of a specific vehicle. [Ref. 4 2 :p. 1421
F. STRUCTURAL BREAKUP OF A REENTRY BODY
The previous sections of this chapter have served to develop the "state-of-the-art,"
as determined through the literature survey, of reentry foimulations and solutions. The
culmination of the uncontrolled reentry process is usually the structural breakup of the
reentry body. The breakup of a reentry body is stated to be a function of surface
temperature, in that, stractural failure Coreakup) is assumed or expected to occur when
the outer structure reaches its melting temperature. [Refs. 63, 86] It has been shown
previously that the maximum temperature a reentry body will experience is determined
by the maximum heating rate, which is a function of ballistic coefficient. [Ref. 86:p. C-
146
11] Although the peak heating rate increases as the ballistic coefficient increases,
ave-rage heating rate (a measure of overall survivability) and maximum local stagnation
region heating (a measure of local "hot spots"), must be considered in determining where
breakup occurs. [Ref. 40:pp. 181-182]
In the 1970's, the Air Force conducted reentry experiments that used optical and
radar techniques to observe actual breakup events. The objective of these tests was to
determine the survivability of reentry body debris. Specific findings from these tests
indicate: .Ref. 86:pp. C-4--C-10]
1. Classical convective heat transfer analysis underestimates the reentry bodysurvivability. Specifically: actual breakup is at an altitude of at least 10 nmlower than predicted; actual surface temperature is at an altitude 10 nrn lowerthan predicted; and the effective heating rate input is a factor of four lower thanpredicted.
2. Consistent catastrophic tailure of mqgnesium/aluminum structures is at analtitude of 42 nm. Figure 42 [Ref. 86] shows convergence of three ballisticcoefficient lines close to the magnesium/aluminum melt zone at this approximatealtitude.
3. Phenomenon of breakup process is independent of: body attitude and rates; body
diameter; body shape; and entry flig!- path angle (0.30 > 1.50).
4. Ballistic coefficient and material of a body determines survivability.
5. A body with low ballistic coefficient will survive reentry, as will a body witha higher melting temperature survive reentry with a higher ballistic coefficient.
6. Reentry body structural integrity is maintained until melting temperature isacheved.
7. Surface structure temperaturte is determined by radiation equilibrium that isbased on a shallow path angle and a low thermai capacity of the outer structure.
147
CD
C\J
z D U)
LI. " Z C\C CO
:D wIJ 0D
m <. z <0- 0
00 0)~
N ~ tMLLI •o
1- 1 LO1
< D t zz ii
in
(5s]L _...*(IhlL NI NC: <JLI
< 0l 0 z
[Ref. 86]I' -
I4I 00
<I 00
77 0I'I0 <
o C/ 0 0 ~
MC LL] 148
Reference [85] concludes that classical convective heat transfer analysis in a free-
molecular flow regime is n,,t indicative of the transition/continuum flow heating which
is responsible for structural breakup. This finding is in agreement with reference [42]
which, as shown previously, discounts the classical reentry heating equations based on
the limitations imposed by the simplifying assumptions. [Ref. 42:pp. 141-142] Reference
[64] also cites the 1970's Air Force studies and ind;cates the deficiencies of the classical
convective heat transfer analysis.
The consistent underestimation of survivability was further investigated in reference
[64]. By using a calibrated heating model, the observed breakup altitude of
approximately 42 nm (79.5 kin) was successfully predicted for satellites with aluminum
structures. Figure 43 [Ref. 64:p. 48] shows the predicted breakup altitudes for two
satellite reentry simulations. [Ref. 64:pp. 7-13]
The heating model was interfaced with a trajectory simulation model in order to
estimate the altitudes where possible breakup events could occur. Table XII shows the
breakup analysis results of the critical elements with their associated material
composition, heating rates and breakup altitudes [Ref. 64 :p. 5 9 ].
Since the breakup analysis was assumed with strictly aerodynamic heating for
structures with low melting points, the authors recommend further investigation for the
combined effect of both aerodynamic heating and aerodynamic loading on breakup.
Specifically, for the case where higher melting point matc rials will survive to al'itudes
where deceleration loads are significant. [Ref. 64:p. 21]
149
4 4
30
20__ = 1_____ IL-L--"
40 380 360 34032030 280 260 240 220 200 180 160 14 120 10080 60 40 20 0
ALTITUDE, 1000 FT
Figure 43: Breakup Altitudes For Corrected Heating Rates
[Ref. 64]
150
Table XII: RESULTS OF BREAKUP ANALYSIS[Ref. 641
Cad Egvrsciie Critical CvA~idate kaqId %A breakup
rfat Costriuction I ALtLttude
peacacraft go"m (I)Fibarglavo 1 12 - to 55Lpoxy
(Z)Katak Htim8r 12 18I 4
tactoIlHeaC Keat radiator (I)K'o 688 69-
Kidiator (UTi 142 69
(333 58 69 28W4C - 69
Aacor/9.aiatin PIadiation Fe I 54. A0 - IU4 JL3-Z
shield shi~a.d
Pressure Vassel Wall (1)Nb 282 165 - M5
(Zkswr-
Full ILA f1 VAIL Nb 282 115 - 1410 A
URI ohu 02 VAIL Nb 282 241 - 1185 b
a Breakup a&1icipaied ror tumbliq~ mode. somepins m~y sunkiv.
b BRmekt anticil tied iu tumbling mode.
151
Another series of investigations of the breakup process are references [10] and [63].
In these studies, the reentry/breakup of Skylab was reconstructed by piecing together all
available data - after the fact. Both of these reports show conclusively, through telemetry
analysis, that the survivability of Skylab was underestimated. Reference [10] cites initial
breakup predictions of 120 km and shows that breakup did not occur until at least 100
km. References [63] and [10] disagree on exactly where the OWS SAS (Orbital
Workshop Solar Array System) separated from the main body, based on telemetry
received at Ascension Island. Reference [10] states that the array was intact over
Ascension Island; however, it was not generating its predicted output. Therefore, it was
concluded that the array was either bent back or physically damaged.
Reference [63] concludes that the OWS SAS was completely separated from thernain body prior to telemetry acquisition at Ascens-on Island [Ref. 63:p- 344]. Finally,
reference [63] postulates a probable breakup scenario as follows: [Ref. 63:p. 344]
1. OWS SAS army (aerodynamically) off at 62 nm / 117 km.
2. ATM separates from remaining OWS at 54 nm / 102 km.
3. ATM SAS arrays separate between 50 and 54 nm / 94.5 and 102 km.
4. ATM and OWS breakup at 42 nm / 77.8 km.
Reference [10] concludes that breakup did not start until some altitude below 100 km.
152
IV. DETERMINISTIC REENTRY/IMPACT PREDICTION METHODS
A. CURRENT REENTRY/IMPACT PREDICTION METHOD
The following section of this chapter describes the various techniques or meth( Is
used by different countries or organizations dealing with reentry and impact prediction
of naturally decaying objects. It is therefore useful to the reader to understand the
current U.S. method used in reentry/impact prediction, since it is the standard by which
all other methods (as determined by this literature survey) are compared.
As stated in previous chapters, the current method for predicting reentry time and
impact location is that of the Space Surveillance Center (SSC) located at Cheyenne
Mountain Air Force Base, or commonly referred to in the literature as NORAD. [Ref.
87] NORAD produces "element sets" which are mean values of the orbital elements that
have been obtained by removing the periodic orbital variations in a particular manner.
In order to use these element sets, and obtain reasonable predictions, these periodic
variations must be "reconstructed" by the prediction model in precisely the same manner
as they were removed by NORAD. Therefore, an input of NORAD element sets into
another model (even though it may be more accurate, or even into a numerical integrator)
will result in degraded predictions unless, as previously stated, the new model can
"reconstruct" the periodic variations. [Ref. 87:p. 1]
NORAD element sets are generated with a general perturbations (GP) model called
SGP4. SGP4 was developed by Ken Cranford in 1970. This mod,'1 was a result of
153
simplification of the more extensive analytical theory of Lane and Cranford (1969) which
uses the solution of Brouwer (1959) for its gravitational model and a power density
function for its atmosphere model. [Ref. 87:p. 3] It should be emph~sized that this
atmosphere "model" is a static representation of density as opposed to the dynamical
models discussed in Chapter II, such as those of Jacchia and others.
The gravitational model includes J2 and J3 harmonics; however, J4 and J5 were
dropped in order to avoid singularities occurring at critical inclination [Ref. 88:p. 2].
Rates of change of mean motion and eccentricity are derived from the density function.
The product of ballistic coefficient and a reference density, denoted B*, is treated as a
solved for parameter. Coupling between J2 and drag is included in the argument of
perigee, right ascension of the node and mean anomaly. The mean motion of SGP4 is
a pure Brouwer, or two-body, mean motion. [Ref. 88:p. 3]
Reentry and impact predictions, in the U.S., are made using a special perturbations
(SP) propagator with conversions between GP and SP theories handled as outlined in
references [87, 89-901. The GP theory is fast, analytical and of low-accuracy, when
compared to the SP theory. SP theory uses a Gauss-Jackson, eighth-order, numerical
integrator, incorporates a 6.12th order geopotential model and applies a dynamical
atmospheric density model (Jacchia-65). The "conversion" between GP and SP theories
is the process of performing an SP differential correction of the initial state vector as
derived from the GP theory (NORAD two-line element set). These are the initial
standards for GP and SP theory compatibility which must be considered in the following
discussion of different reentry/impact prediction methods. [Ref. 9 1:p. 8]
154
The first TIP (Tracking and Impact Prediction) run is performed approximately 15-
20 days prior to the estimated reentry date, which is initially predicted by the GP model.
This is also when tasking of the observation sites is initially increased in order to support
high accuracy SP processing. [Ref. 9 2 :p. 1]
Orbit determination is accomplished through a first-order, linear, weighted, least
square, curve fitting process, commonly called differential correction. Sliding fits are
used to process both new and old metric data until the satellite is "no longer in orbit."
The force models used are the Jacchia dynamic atmosphere (1965) and the World
Geodetic System Earth gravity model (1972). The Earth gravity model is truncated to
the sixth order for satellite decay predictions. [Ref. 92:p. 2]
Direct step-by-step numerical integration of the total acceleration acting on the
decaying satellite is accomplished in the manner of Cowell's method. Gauss-Jackson
eighth order predictor-corrector formulas, in ordinate form, are used to integrate the
equations of motion. Because of computer run-time constraints, the partial derivatives
necessary for differential correction are computed analytically except for the secular
variations due to atmospheric 6,ag of the orbit semi-major axis and eccentricity. These
parameters are integrated numerically using a low-order (trapezoidal rule) integrator.
[Ref. 92:p. 2]
155
Currently, the differential correction solution state consists of the equinoctial
elements and satellite ballistic coefficient model parameter, B0. After each solution, the
new state is used to predict a decay time (when altitude equals 10 km). [Ref. 92 :p. 2]
B. ALTERNATE REENTRY/IMPACT PREDICTION METHODS
1. Reentry Prediction Methods At ESOC
One method used at the European Space Operations Center (ESOC) for the
prediction of reentry and impact of decaying satellites is an improved and computerized
version of the King-Hele technique, reference [55]. [Ref. 93] AnotheT method used at
ESOC is based in a computer program called FOCUS [Ref. 94].
The principal characteristic of FOCUS is its ability to overcome the
deficiencies of the semi-analytical orbit prediction techniques at low altitur'es [Ref. 94:p.
26]. Recalt that in near-Ear-th orbits, 200-700 km, Earth oblateness (J2) is regarded as
the only first-order perturbation. Higher zonal harmonics and air drag are regarded as
second-order contributions. All other effects are considered as less than second-order
perturbations (less than J22 in magnitude). [Ref. 94:p. 25] Eventually, the satellite passes
through an altitude regime where the air drag force (caused by increasing atmospheric
density) reaches a magnitude of the same order as the J2 Earth oblateness effect. At this
altitude, - 150 kn., depending on the vehicle's ballistic coefficient, the accuracy of
analytically derived drag perturbation results strongly deteriorates. [Ref. 94:p. 26]
FOCUS stops the state propagation after passing through a user-defined
altitude shell (h = 170 km specifically for the case of Salyut-7/Cosmos-1686, which is
156
the focus of reference [94]) and forwards a calculated osculating Keplerian state vector
at epoch, together with all relevant perturbation parameters. This new set of data is then
numerically integrated and the reentry trajectory is propagated until shortly before impact
with the Earth. [Ref. 94:p. 26]
Several key features of the FOCUS program are: [Ref. 94:p. 26]
1. Perturbatlon equations:
(a) Cowell's formulation of the perturbed Newtonian equations written in termsof six first-order, differential equations for each component of the Cartesianstate vector
(b) Reference frame is the mean equatorial system of date
2. Perturbation models:
(a) Geopotential model GEM 10B (J 2-J 7 used)
(b) Atmospheric density models: MSIS-86 for altitudes > 120 km, U.S.Standard Atmosphere (USSA-76) for altitudes :5 90 kim, and a bridgingfunction for altitudes between 90 and 120 km.
(c) Variable drag coefficient, CD = f(Ma, Re, Kn)
(d) Co-rotating Earth atmosphere
(f) Luni-solar third body attraction (point mass)
(g) Solar radiation pressure
3. Integrator:
(a) Runge-Kutta/Shanks 7/8 single step method for generation of a starting arc
(b) Adams-Bash forth/Adams-Moulton(AB/ AM) forth-orderpredictor-corrector,multi-step method for propagation of the Cartesian state vector.
157
(c) Non-regularized time (t) used as an integration variable, with constant stepsizes of At = 30 sec.
A significant limiting feature of this program is that the reentry trajectory is
terminated at 30 km altitude because the governing laws of perturbed Keplerian motion
become invalid below this threshold altitude. This criterion is marked by a decrease in
the orbital energy to a level, where the aerodynamic forces are essentially in balance with
the zero-th order central gravitational attraction term. [Ref. 94:p. 26]
The reentry vehicle is, however, considered to be in nearly vertical fall from
an altitude of 30 km and below. Thus, there is only a minor dispersion of the impact
point during the final seconds of flight. This rationale leads to the conclusion that it is
not necessary to perform another transition from strongly perturbed Keplerian motion to
an aerodynamic flight phase for the integration to Earth impact. Thus the Center Of
Impact Window (COIW) is defined as the location at which the vehicle passes through
the 30 km altitude. [Ref. 94:p. 26]
The aerodynamic transition regime is defined as a computed, weighted mean
of PMsIs and PUSSA where
PMSs =f (h, 0, X, UT, t, t4, •.7, F10.7, A ,) (153)
P usS4 = f(h) (154)
which results in
158
P = wmsas Pusts + (1 wMvs) P us (155)
where the weighting factor
wu,,t(h) c [0,1] (156)
is defined for the altitude region h120 Ž h : hg0 as
1c t ( -- _cos 1+ (157)2 hi2o-h9) 2
where
h = geodetic altitude
4P = geodetic latitude
X = geographic longitude
r = local solar time
UT = universal time
td = day of the year
Whe1 a the altitude is high enough to maintain
Kn. d> 1 (158).Vic
where
dv/ = characteristic length of vehicle
159
then CD is equal to a constant given by CoK, provided Kn, > 1. As the altitude
decreases, a transition to hypersonic continuum flow is entered, where Kn0, = f(h) and
[Ref. 94:p. 27]
CD = c1 + Cj'log,(c•'Kn,,) (1n9)
provided that 0.02 :r Kn,, _5 1. During the hypersonic continuum phase another
constant CD level is attained (about 50% less than the free molecular flow value), which
is given by
CD c = CO"a (160)
provided
Kn,. _ 0.02
"Thie next phase of supersonic and transonic continuum flow can be
approximated by an altitude dependent Mach number, Ma = f(h), where
CD +CJ' Ma (Ma. - exp [ Ma (M 4)(161)
provided
KnQ. Ž 0.02
0.8 S Ma0. s 5
Finally, in the subsonic phase, the drag coefficient is dependent upon viscous
interactions and therefore Reynolds number, Re = f(h), given by
160
C, = C' (Re.) C*c' (162)
provided'
Kn. < 0.02
Ma•. 5 0.8
These dependencies, CD = f(Kn, Re, Ma), with the underlying altitude
functions Kr(h), Ma(h) and Re(h) are from the U.S. Standard Atmosphere (US$A) and
are incorporated into the numerical ree. t .y prediction softwz-e. These model constants
are limited, however, to spherical and cylindrical shapes, with their longitudinal axis
perpendicu!, to :he airflow. [Ref. 9 4 :p. 27]
Since Ma = VL / V.d and Re = V., d/v are functions of geodetic latitude
related by Vwud(h), kinematic viscosity, v(h), and aerodynamic velocity, V,,., the free
fall of the reen:tering vehicle in the lower atmosphere may be determined by iteratively
solving the fohowing equation for the equilibrium descent velocity, V,.,
v h2 = - 2M_ g(h) 63VI; ~h' 2 - _____ (163)
j p(h) cD (h, v1§)
where
g(h) = central gravitational acceleration of the Earth
p(h) local air density
Using this technique, in the post-flight analysis of Salyut-7/Cosmnos- 1086, the
following values are determined:
161
m/A = 159.5 kg/m2 "
therefore V.,, at
h = Z5 ion, is equal to 330 m/s
h = 20 km, is equal to 100 m/s
h = 10 km, is equal to 70 m/s
Figure 44 shows the ESOC reconstruction of the Salyut-7/Cosmos-1686 final
descent altitude profile. The actual reentry occurred at 0345 UCT on 7 Feb. 1991 over
(39.3°S, 69.7°W) [Ref. 94:p. 29]. Table 13 shows a comparison of reentry predictions
from the U.S. and various European and former Soviet sites [Ref. 94:p. 32].
The conclu-ions of this European Space Agency (ESA)/ESCC report indicate
that there was good correlation between ESA, U.S. and former Soviet predictions. There
was some difficuiy in interpreting the Soviet orbit determiiiiiatiozi resuiil -b 4ppdi AiI ,y
12 hours prior to reentry. Also at this time, the U.S. elemv.nt sets became "time late,"
due to transmission delay times from the U.S. to ESOC. Therefore, the ESA/ESOC
predictions could not be maintained in "real i:me" after this point. [Ref. 94:p. 33]
2. The LIFETIME Model
The Aerospace Corporation previously developed the LIFETIME program
and it was further refined through the work of reference [95]. This program is similar
in some respects ;.,j the previously discussed ESA program, FOCUS. This program
offers a "fast, efficient computer tcol for orbital lifetime estimation" [Ref. 95:p. 17].
An advertised major benefit ot the LIFETIME program is its usefulness as a "highly
accurate, real-time reentry prediction tool." [Ref. 95:p. 20] This may also have
162
Eu
awiil4J l
Figure ~ ~ ~ ~ ~ ~ ~ I 44ccyt7Csos18 ia ecntAttd rfl
1631
Table XIII: SALYUT-7ICOSMOS- 1686 FINAL DECAY PREDICTIONS[Ref. 941
Re-Entry PredictionsPrediction (times In UTC)
Source Rcoived Re-Entry Prod.at ESOC (Col")
NASA GSFC 01-Feb 14.58 08-Feb 21:33ESOC MAM 02-Feb 0)4:27 06-Feb 22:30C NES CST 03-Feb 11:33 06-Feb 04:00ESOC MAS 04-Feb, 10:30 07-Feb 01:37
N4ASA CSFC 04.Feb 16:04 07-Feb 03:38CNUCE CNR 04-Feb 19:43 07-Feb 03:17CNES CST 05-Feb 0A427 Oe.Feb 19:23ESOC MAS 05-Feb 10:00 Q7-F-h 02:23
NASA GSFC 05.-Fwb 15:05 07-Feb 03:05ESOC MAS 06-Feb 11:33 07-Feb 03:50RAE 08-Feb 11:38 07-Feb 04:30
CNIJCE CNR 06-Feb 12:50 06-Feb 22:26
NASA GSFC 06-Feb 14:09 07-reb 03:28CNUCE CN__ 06.Feb 16:51 07-Feb 01:24
NASA GSFC I06-Feb 15:56 01-Feb u)414
CNES C$T 06-Feb 17ý00 07-Feb, 05.00ESOC MAS 06-Feb 19.41 06-Fob 04:19CNUCF CNR'- 06-Feb 22:05 07-Feb 04:43
CHES CST 06-Feb 22:15 O7-Fob 04:38
IKI lAM 0$-Feb 23:37 07-Feb 0.4:40
NASA GSFC 06-Feb 23441 07-Feb 04.05
C SOC MAS 07-Feb 01:00 07-Feb 04 5011(1 lAM 07-Feb 0 1.02 07-Feb 04:38
11(1 MCC 07. reb 0 1:02 07-Feb 04-20
CNUCE CHR 07-Feb 01:07 07-Feb 04:00
IKI MoD 07-Feb 01:57 07-Feb 04:00
IlKl MCC o7-Feb 02:22 07-Feb 03:57IKIi 1AM 07-Feb 0.4.00 07-Fet 03:37ESOC MAS 07-Feb 04-20 07-Feb 04:29
IKI MoD' 07-Feb 04.34 j07-Feb 03.47
E 5OC IN S 07-Feb l 100-Fb0.5
164
application as a first time modeling tool as compared to methods capable only of post
flight analysis of reentry events.
The LIFETIME program uses either the Jacchia-Walker (64) or Jacchia (71)
atmospheric density models for altitudes above 90 km and uses the U.S. Standard
Atmosphere (1962) for altitudes below 90 km. The latest version (4.0), the subject of
reference [93], also allows differential corrections of the ballistic coefficient and
movement of solar panels to simulate sun tracking. [Ref. 95:pp. 18-19] Version 4.0 was
designed to solve a major deficiency of version 3.0, in that, there was a built-in
uncertainty in Earth impact time prediction of at least one orbital revolution. This was
inherent due to the minimum step size of one revolution, which was a function of using
the averaged equations of motion. [Ref. 95:p. 22]
A "unique" feature of the LIFETIME program is the differential correction
of the ballistic coefficient. By using the least squares method, the following equation for
differentially correcting the inverse ballistic coefficient (B*) can be formulated [Ref.
95:pp. 28-29]
d al i + Ae eLEN, A I - IO. 14
AB* aB" - (164)ad,- ) 2] , a ,I
where
N = number of observations
• a =/'a,,
165
a, = observed semi-major axis at time i
a0 = semi-major axis at epoch
ei = observed eccentricity at time i
It can be shown that orbit decay in semi-major axis and eccentricity is
directly proportional to the product of inverse ballistic coefficient and density (B*p).
This differential correction process (which absorbs the atmospheric density uncertainties)
results in a "converged ballistic coefficient" through multiple iterations of equation 175.
This is how LIFETIME becomes more accurate than the atmosphere model it uses. URef.
95:p. 29]
LIFETIME integrates the averaged equations of motion using a step size of
multiples of the orbital period, until the satellite reaches a specified or default altitude.
Based on [86], the default altitude chosen is 42 nm (78 kin) and this is where "breakup"
is said to occur. LIFETIME then backs up to the last propagation step, resets the orbital
elements and time to the previous step's values. Next, the Runge-Kutta 7(8) integration
routine is invoked and the satellite's orbit is propagated through breakup to impact. tThe
elements are converted from classical mean to classical osculating elements and finally
to Cartesian elements during the numerical integration process.) [Ref. 95:pp. 32-601
Numerical integration is the preferred method for dealing with the final stages of reentry
to impact since it is the "most efficient and accur,.te way to handle regions where a high
rate of changc in the equations of motion" are prevalent [Ref. 95 :p. 55].
166
Several significant user input altitude values are: [Ref. 95:p. 59]
1. RKALT - lower limit at which LIFETIME converts to Cartesian numericalintegration. Default = 0.000 kim, most effective if > BRKALT.
2. BRKALT - vehicle breakup altitude. Default = 77.784 km.
3. ENDAL.T - perigee altitude lower limit, ends integration and propagation.Default = 10.000 kin, this should be set to 0.000 km to model impact.
At the point where the satellite reaches the BRKALT, the lat/long projection
of the satellite is recorded. This point is defined as the "debris heel point," or the first
point along the gxoundtrack where debris could potentially impact. Propagation continues
unti! ENDALT, where the lat/long is identified as the center of mass impact point.
LIFETIME then resets the Cartesian elements to those at the BRKALT point, however,
B is changed to 60 lb/fIW and the propagaton to ENDALT is continued. This allows the
computation of a "debris toe point," or the point of farthest travel of any breakup debris.
The value of 60 lb/ft2 for the ballistic coefficient is based on reference [86]. [Ref. 95:pp.
39-60] Figure 45 shows the debris dispersion footprint as described above [Ref. 9 5 :p.
60]. Figure 46 shows the LIFETIME groundtrack and impact area plot as well as a
sample altitude decay hiistory [Ref. 95:p. 631. (An item of interest not addressed by the
author of reference [95] is the fact that th" center of mass, in this sample plot, impacts
after the "final re"bris impact. ")
The results of impact prediction accuracy comparisons between LIFETIME
3.0 and 4.0 do not definitively show 4.0 as a significant improvement over 3.0.
Howevcr, some notable improvement in accuracy for version 4.0 can be shown and
107
Cl
I-
Figure 45: LIFETIME Debris Dispersion Footprint[Ref. 95]
168
*: • - :.. ,.u7I •NS Pill'•
-". flft Uniflierm
"" ' . L - A
-60.90
-90 ______________________________--_________
-180. -1,o. -120. -. 0 -60.0 -30.0 0.0 30.0 60.0 90.0 120. 15C. I80.LONGITUDE IDEGr
120. f - ___-_-
120o .o .. .... .. ...................... ................ ...... ...................... ......... ..................................†.. . .. .....................
40.0 ... ~ K .W ........... ........... 7................... .... ... ....... ........ ........
0.00 0.15 0.30 0.15 0.60 0.75 0.90 1.05 1.20TIME (HOI•RSI FROM 1992/ 5/24 AT 14: 3:30.589 GMT
Figure 46: LIFETIME Grouidtrack And Altitude Decay History[Rc~f. 95]
169
greater confidence is held in version 4.0's period calculations and final impact
predictions. [Ref. 95:pp. 65-70]
A sensitivity study and analysis was performed in order to determine the:
1. Effects of NORAD data span on program accuracy
2. Effects of prediction span on program accuracy
3. Effects of solar conditions during differential corrections
4. Effects of solar flux on overall program sensitivity
The results of the sensitivity study and analysis are as follows: [Ref. 95:pp. 89-94]
1. NORAD data span:
(a) Period calculation accuracy - no strong trend for sen. itivity noted
(h) Impact predietion accuracy - general trend for less averaged impact timeerror for greatei time spans
2. Prediction span - impact time errors show a strong trend for increased accuracywith shorter prediction spans.
3. Solar flux - the results indicated a complex relationship between the estimatedsolar flux for the prediction period, the actual values of solar flux, theprediction span, and the resulting impact errors.
A general observation and conclusion is that fairly steady solar activity at
modest levels provide for the most accurate impact prediction, especially when using the
solar flux value of the last day on NORAD data as a constant during the prediction span.
During pefiods of highly variable solar flux, the last day's value of NORAD data may
170
not give the least error, however, this conservative assumption will not result in the
maximum error either. [Ref. 9 5 :p. 92]
An appropriate concluding remark of reference [95] is that
It seems there are still dynamic aspects of the final orbit decay and reentry processthat warrant further study. [Ref. 9 5 :p. 93]
3. Modeling Ballistic Coefficient
Analysis of historical data shows that in the final days prior to decay, the
ballistic coefficient, B, (CDA/m) exhibits a pronounced variation with time. This
variation is due to one or more of several causes: [Ret. 92 :p. 2]
1. Tumbling or weather vaning of the satellite
2. Loss of mass
.)..•a V "UtdLIUl| UI Ui,,, c • U i.iLi.L. ,
Presently, none of these effects are modeied explicitly since this requires precise
knowledge of the multiple variables described in Chapters II and IIl, such as vehicle
mass, attitude and shape. Additionally, any unmodeled forces acting in the in-track
direction, such as those inaccuracies in the atmospheric density model, can manifest
themselves in the solution ballistic coefficient. [Ref. 92:p. 2]
Despite the lack of specific knowledge of unknown forces acting on a
decaying satellite, an improvement in reentry prediction accuracy can be made by
introducing a new model parameter. This parameter takes into account the fact that for
many decaying satellites, the variation in B appears to be linear during the last several
171
days prior to reentry. This new parameter is B0-dot, the time derivative of the ballistic
coefficient, which is parameterized as
B = BO + ho( t)(165)
where subscript 0 indicates the quantities at the solution state epoch, to. [Ref. 9 2 :p. 3]
The introduction of Bo-dot requires that new semi-analytic partial derivatives
for B0 and B0-dot be calculated. These must be determined over a finite time interval
using a differential correction fitting process. Several such differential corrections must
be made over the course of the final days of orbit decay and reentry, with a new Bo and
Bo-dot being determined for each differential correction. It is necessary to start the
fitting process with close approximations of B0 and B0-dot for the first differential
correction. These initial "guesses" come from analysis of historical data. [Ref. 92:p. 3]
Reference [92] studied 264 reentry events over the time span from Jan. 1977
to Mar. 1979. All of these objects were processed as TIP reentries and historical records
from the SSC were used in this investigation. In order to optimize the data, the rercords
of 156 reentries were chosen, since these represented rocket bodies and payloads from
the former USSR, which presumably would have well-known physical attributes. Table
14 shows the data distribution used throughout this investigation. [Ref. 92:pp. 3-4]
The development of the solutions for the partial derivatives of B3 are not
shown here for the sake of brevity, however, the results are as follows [Ref. 92:pp. 10-
172
Table XIV: DISTRIBUTION OF DECAYED ORBITAL OBJECTS[Ref. 92J
DISTRIBUTION OF 264 TIP LOG DECAYED ORBITAL OBJECTS
YEAR- 1970mohrmL J F M A M Ii A SO0N DNUmE1~t.' 2
YBLL -. --1977MONTIL I F M A M I I A SO0 N DNJMMU.' 8 11it0 15 8 8 11 14 9 8 7
YEAW. 1978MO=.-: I F M A V.I JA SO0N DN(JMdEER it 19 913 612 7 11it131818
YEAR; 1979MON'mt I F M A MJ I A SO0 N 1NUMUL. 6 10 7
YEAR. 1983MONTH: J-V FMA M I J A 50 N DNUNB2.' 1
*Sele~cted 156 Soviet orbitally decayed oloject diStribution:O.7,3
ROCKET BODIES DIA(nm) LNG (mn)21 A2 VOSTOK (440 kg) 2.6 7.589 A2 VOS[T),C (2500 kg) 2.0 2.0
4 81 como (1500 kg) 1.65 8.08 Cl SKEAN (2000 kg) 2.0 6.03 IN PROTON (1900 kg) 3.9 4.03 Dl PROTON (4000 kg) 4.0 12.03 FIM SCARP ( 700 kg) - 2.0 - 5.01 FI1M SCARP (1500 kg) 2.5 6.0
I'AYLOADSI3 NAVSAT (875 kg) ?
1 (325 .kg)1(3640 -kg)I -*( 680jkg)
a * MILSAT. ( 400_ kg)I MIL/SCi (' 50, kg)4 RORSAV4 (4500 igY'2 MIL ( 900 k9)1 (11,20 kg)
CThe exact size iwd shape of tho payloads arc unknown at this POin~t.
173
B2t aB ,, .vF (166)
0 Bo to" B e
Bo &o=B ago ( I dt (167)
BO L- = 3 n°1 (t to) f lo 1 6dt (169)--aBO O 4 o a
BO -E _ Ro ,9• = 0 (170)
where
af = e cos (0) + •)
a, = e sin ( of + c) oc tu
S= time rate of change of eccentricity due to drag perturbations
h. = time rate of change of serni-major axis due to drag perturbations ,
L = mean longitude
X = tan (i/2) sin f
' =' tan (i/2) cos f
174
Similarly, the partial derivatives of B0-dot are givei [Ref. 92:p. 12]
--BoZ =-Aa, 3 0[ 0 1- dt (171)
"ao a- oaft['-o ) 1 Be
S-B dt (172)
3 * :[ 1h _ B=oasof (t--to) .i dt (173)
_a.o 2 B
6( az.o _ 3 j. jo (t-to) (t-to) 1 61 dt (174)a,60 4 a0 [to B (1)
a60 ao 0 (175)
Of the 156 cases studied in the development of the new parameter and its
associated partial derivatives, two case studies were chosen for the application of the new
method. These cases were Cosmos-954 and Cosmos-1402, both of them being significant
due to their potentially survivable nuclear reactors. [Ref. 92:p. 13]
The conclusion of this paper is that the new method is more accurate than the
conventional method as evidenced by the comparison of the best fit differential correction
root mean square (RMS) of residuals for each of the 25 sliding-fit runs over the last nine
days of orbit for Cosmos-954. The same comparison is made for Cosmos-1402 with the
175
only exception that 26 vice 25 SP differential corrections were run. The results of these
comparisons are shown in Figure 47 [Ref. 92:pp. 15, 17].
As stated previously, the historical data analysis of most satellites showed the
linear trend in B only over the last 3-5 days prior to reentry. However, one of the two
specific cases chosen for application of the new method, Cosmos-954, did not exhibit this
trend until the final day prior to reentry. Table 15 shows a comparison of decay time
error root mean square over the last five days of orbit (for these two case studies) as well
as a similar comparison aftzr the linear trend in B becomes apparent in both reentries.
[Ref. 92:p. 18]
Another method using a correction of the ballistic coefficient comes from
reference (47]. This investigation states that during the jeint international effort to
r•,di;, thC MA-ntri mn~rt Ot S1uut-7/Cosmnis- 186, it was ohbserved that the "most exact
data" had consistently been presented by the Space Control System (SCS) of the former
USSR. This was explained by the fact that:
... great attention has been given to the problem of determination and the predictionof the satellite motion in the atmosphere: for the solution of this problem they havebeen conducting the cycle of theoretical and experimental studies during [the last]20 years. [Ref. 4 7 :p. 35]
The model used by the SCS to describe satellite motion will, to a great
extent, determine the "completeness" and precision of the time and area of low-orbit
satellite reentxy predictions. Gravitational field models in spherical functions of 36 or
higher orders are not necessary, they only serve to increase the computation time. At
176
.3.00
2.75
2.50 P Bo SOLTlON,. ,., s3 .Dt. r SOLUTIU.N
2.25
2.00
" 1' 1.75v1.50
S1.25
1.00
0.75
0.50
0.25
0.00-,I .
-0.0o -So -7.0 -,d.ýo -'_ __ -zo -- ".0"TIME TO DK (DAYS)
1 .00 1
O.00 ,' • X, IO SXl UTION
f0.6 0
0.40
0.20 -.
10
-9.0 -8.6YT'116 i~ -C .0 -40 -Lid-1.0 -1.3 0.0TIME -0 TDK (DAYS)
Figure 47: Best Fit DC RMS For Cosmos-954/1402[Ref. 92]
177
Table XV: DECAY TIME ERROR AMS[Ref. 921
DECAY TIME ERROR ROOT MEAN SQUARE (RMS) OVER LAST 5 DAYS
Decay Error RMS(hrs)
Decay Error RMS(hrs) Bo, Bo Solulicn,
Satellite B] Solution J30--O.0 for Prediction Improvement(%)
COSMOS954 3.51 4.11 -17.09
COSMOS 1402 1.59 1.43 10.06
IDECAY TIMIN vilmR ROOT MEAN SQUARE (Rt$) AFT LI1 B UECOM•EIS L•.tW:,i-
Decay Error RMS3(hrs)
Decay Error RMS(hrs) Bo, o30 Solution,
Satellite Bo Solution !3o-0.0 for Prediction Improvement(%)
COSMOS 954 1.22 0.48 60.66
COSMOS 1402 1.51 0.98 35.10
178
low altitudes, two to three days prior to reentry, a simple geopotentiai mode! is suitable,
since it is the atmospheric resistance force which is increasing by several orders of
magnitude, not gravitational forces. [Ref. 47:p. 36]
Predici'on of low-orbit satellite motion is accxmplished by numerical and
semi-analytical integration methods. The semi-analytical methods are based on the
asymptotic solutions given by Krilov-Bogoljubov, Zeipel-Brouwer averaging methods.
[i<eferences given are written in Russian and the authors wer,. unable to obtain English
translations.] [Ref. 47:p. 36]
The actual method applied to a reentry event may be a variant of some
numerical and semi-analytical methods. These specialized fast-acting, semi-analytical
prediction algorithms for low-altitude orbits have "counting times of order 0.01 seconds"
for a 24 hour prediction span. [Ref. 4 7 :p. 36)
Th.. reentry of the Salyut-7 complex was predicted and confirmed by
observation in the range of 75-105 km. In this altitude range, it was also observed and
predicted that breakup would occur. [Ref. 47:p. 39)
It was assumed that the ballistic coefficient would vary during reentry, to the
same degree as it did prior to an attitude control maneuver. This assumption allowed the
reentry problem to be solved by varying the initial conditions within their ranges of
possible errors, with subsequent integration of motion equations up to the point of
reentry. In many cases it was only the ballistic coefficient which must be varied. This
sensitivity of reentry time to the ballistic coefficient is given by the following
approximate formula [Ref. 47 :p. 39]
179
=Wk (176)
where
= time interval of motion prediction to the moment of reentry
k = ballistic coefficient = CDA/2m
Given a 1% change in ballistic coefficient, the down range travel is altered
by 180 km. A 2-3 % change in ballistic coefficient (as seen prior to the attempted control
maneuver) alters the predicted impact point by ± 500 krm. Figure 48 shows tLe
dependence of time, latitude and longitude on the ballistic coefficient [Ref. 4 7 :p. 40].
The approximatc estimate of the probable debris fallout range, considering
the errors in determining the reentry point, with the vehicie destruction occurring at 75
k~m alttude, is equa] to ± 1000 kmin relative to the c.alculated impact point. This
corresponds to the sub-satellite track crossing Chili and Argentina from the south-west
to north-east, [Ref. 47:p. 40]
180
03.48 3w50 HOP
03.43' N v~
03,*38______
Fligure 48: Dependence Of Time, Latitude- Arid Longitude On B[Ref. 47]
181
C. FACTORS INFLUENCING REENTRY/IMPACT DISPERSION
The reentry vehicle's deceleration may be expressed in terms of "gravitational
acceleration forces" (g's) as follows [Ref. 96:p. 1]
1 V2
Dynamic pressure 2 (177)Ballis-ic coefficient W A
CD
Assuming that density and wind velocity are the two atmospheric characteistics
which most strongly influence the motion of a reentry vehicle passing through it,
reference [96] states the following:
1. Reentry vehicle is non--ablative and non-maneuvering (NV and A are constant)
2. Drag coefficient is dependent upon altitude (Mach dependcice)
3. Drag coefficient is shape dependent
The displacement in impact point can be considered to be the sum of miss
contributions from the various atmospheric layers, given by the product of influence
coefficients times changes in density or wind velocity for the respective altitude layers.
[Ref. 96:p. 2]
Reference [97] studies the effects of a non-spherically symmetric atmosphere on
the reentry/impact location of a decaying satellite orbit. This investigation deals with two
specific cases, a 90 degree and 30 degree inclined, low Earth orbit and shows the effect
of the atmosphere's diurnal bulge on the impact location. It is observed that a decaying
182
object is "more likely" to reenter and impact the Earth at certain latitudes than at others.
[Ref. 9 7 :p. iii]
The principal effects contributing to this impact distribution, in addition to the
diurnal bulge, are the geometric oblatenss of the atmosphere and the gravitational
oblateness of the Earth. Also identified as important parameters, are arbital inclination
and ballistic coefficient.
The results of this investigation are as follows: [Ref. 9 7 :p. 43]
1. Polar orbit (90' inclination) reentry is most likely in the equatorial latitudes.
2. Latitudes of maximum or minimum likelihood for impact are not significantlyaffected by changes in ballistic coefficient.
3. Large, "balloon type" (spherical) satellites react to the diurnal bulge by causingimpact on the opnosite side of the Earth.
4. Smaller, heavier satellites show little influence of the diurnal bulge, but theyexhibit a greater variation in impact probability with latitude.
5. The impact distribution for a given spherical satellite "flattens" as the inclinationis moved from 90° to 30'. This is attributed to the fact that the extremes inEarth radius differ by only 5 km under the 30 ° inclined orbit as opposed to 21km under the polar orbit.
Reference 198], a survey of satellite lifetime and orbit decay prediction, conducted
in 1980, states:
-gravity perturbations cannot, by themselves, lead to orbital decay since they areconservative. But, they can induce oscillations in the orbit... drag is proportionalto atmospheric density.. .gravity perturbations coupled with drag are moresignificant than when only drag is modeled. [Ref. 9 8 :p. 3]
183
Also noted is the observation that non-symmetric mass distribution and shape may
amplify the torques created by: [Ref. 98:p. 41
1. Gravity gradients
2. Aerodynamic forces
3. Solar pressure forces
An interesting note, also, is that any mass loss increases the magnitude of the drag
deceleration since [Ref. 98:p. 5]
1
a,• - (178)
A final noteworthy observation from reference [98] is that any attempt to model
satellite attitude, shape, lift, mass distribution, ablation and breakup would largely exceed
the level of sophistication currently available (1980) for atmospheric density models, -
therefore, these non-atmospheric factors are neglected and treated as higher-order effects.
[Ref. 98:p. 5]
Reference [99] was initiated after the reentry of part of Sputnik IV over Wisconsin
in September 1962. [Ref. 99:p. 2] This study of factors influencing the prediction of
orbit decay and impact states:
S...final reentry occurs shortly after the point of minimum altitude (not coincidtntwith perigee, because of the Earth's oblateness) [Ref. 99:p. 29]
It is possible, knowing only the initial inclination angle and reentry vehicle orbital
path, to determine the points at which the minimum altitude points will occur.
184
Therefore, knowing where these points occur, it may be possible to predict reentry within
seweral tenths of an orbit revolution, as opposed to ambil uities of one complete
revolution as noted in reference [951 (LIFETIME v30), since reentry tends to follow the
point of minimum altitude. [Ref. 9 9 :p. 33]
Without such precise knowledge of the body's ballistic coefficient, the
characteristics of the atmosphere, the orbital elements and their variation, only
"probability estimates" can be made of impact latitude, based simply on the relative time
spent by the satellite in various latitude bands. This conclusion yields the following
relationships: [Ref. 99:p. 33]
1. If the initial inclination equals 500 - three times more likely to impact between40-.5U0 band than 0-10' latitude band
2. If the initial inclination equals 490 - two times more likely to impact thecontinental U.S. that a 65-" inciined erbit
A final comment about factors influencing reentry and impact prediction from this
inveot-ation is that:
In spite of the present reentry rate of about one object a day (most of them burningup, and 2 out of 3 of Russian origin), the threat imposed on the Earth populationfrom direct hits by debris from uncontrolled reentries of space objects cangenerally be regarded as minor when compared with other risks of civilization(traffic accidents, etc.). An accumulation of worst case assumptions could lead toone casualty every five years for a densely populated area of the size of the FederalRepublic of Germany in case of no prior warning. In case an early warning couldbe issued, such casualties could most likely be avoided completely. [Ref. 100:p.49]
Figure 49 shows the magnitude of the low Earth orbit problem purely as a function
of the number of objects in orbit [Ref. 100:p. 51]. Figure 50 shows the limiting factor
185
* . *4 *. *
*6 d
C. r
[Ref. 100]S
186. *I *
Radar Anda Assets
Fiue5:USalectro-optical Sesr^8t
(ef.s
o18
of geographic location of observation sites from which data may be obtained for
reentry/impact predictions [Ref. 10 0:pp. 5-6].
The final factor in reentry/impact prediction to be discussed here is the rotation of
the orbital plane during the final phase of reentry, as presented in reference [101]. This
study states that planar motion is representative of, or an adequate model for, reentry
deceleration and heating rate studies. It is, however, inadequate for the study of ballistic
reentry below altitudes of 60 km and ballistic coefficients in the ranges 0.001 md/kg <
B < 0. 1 m2/kg. For these values of B, the orbital plane begins to rotate at altitudes of
30 km and 60 km respectively. [Ref. 101:p. 1215] This investigation deals exclusively
with the final descent from 120 kin, and defines this as the "reentry phase" [Ref. 101:p.
12161.
As previously mentioned, maximum heating rates and deceleration occur at altitudes
high enough, such that, rotation of the orbital plane is negligible in the calculation of
these parameters. However, in the calculation of the total range traveled to Earth
impact, these lower altitude effects and orbital plane rotations become significant. [Ref.
101:p. 1217]
Several initial assumptions are made: [Ref. 101:p. 1217]
1. 0.001 m2ikg r. CDA/m s: 0.1 m'/kg
2. 0 .5 *0' :. 450 , where it0 is the initial reentry trajectory angle
3. v, < 15 km/sec
188
As the vehicle descends, the ambient air becomes denser aid the rotating
atmosphere begins to force the vehicle's motion towards a direction parallel to that of the
atmosphere's velocity. This effectively rotates the vehicle's orbital plane towards the
direction of the atmospheric velocity vector. Figure 51 shows this rotational effect on
the orbital plane [Ref. 101:p. 1219].
,•-axis points to the north. r -axis to the .cast, the plane 7 -is the surf'ace of the 'inertial Earth'1•- •
Figure 51: Orbital Plane Rotation Due To A Rotating Atmosphere__[Ref. 101]
189-
_r=.
IIai pitst henrh.1-xi oth IItcpln Is th su c of thi ineta I at
V. STOCHASTIC AND STATISTICAL PREDICTION METHODS
This chapter will describe three investigations that used stochastic and statistical
techniques and methods to predict the time and impact location of decay-induced
reentry. Two investigations have direct applicability to the current reentry model used
by the USSPACECOM and the third investigation is a Monte Carlo simulation analysis
that was used to predict the footprint dispersion area of Skylab.
A. EXTENSIONS OF THE PHYSICAL MODELING REENTRY THEORY
In Chapters II and IV, the current reentry theory used by the USSPACECOM to
predict impact location and time was presented. The theory's accuracy is limited by the
ability to observe the reentry process and the inherent deficiencies in the model used to
represent the physical reentry process. This section will examine a Doctoral dissertation
and a Master's thesis from the Air Force Institute of Technology (AFIT) which are
proposed extensions or modifications of the current model. [Refs. 48,102] These
investigations utilize stochastic and statistical methods and techniques to improve the
predicted impact location and time.
1. Estimation of Recttry Trajectories
Reference [102] developed a linearized differential corrector technique as an
extension of the existing orbital estimation technique into the reentry region to estimate
the reentry trajectory. Reentry observations from a space based sensor capable of
190
providing infrared angular trajectory observations at fixed, discrete time intervals and
large uncertainties in true reentry dynamics, were the primary engineering considerations
of this problem [Ref. 102:p. 1]. The author states the following about the current
deterministic model:
... the basic differential corrector with deterministic dynamics is inadequate forprocessing true reentry data. The deterministic dynamics and infinite memory ofthe basic formulation, cause the estimator to yield significantly biased solutionsrelative to the standard deviations of the estimator-computed covariance matrix.These occur when processing data reflecting the dynamic variations anticipated intrue reentry trajectories. The use of the estimator in this form would not providean accurate estimate of Earth impacts for satellite debris. [Ref. 102 :p. 19]
Since the uncertain dynamics of the reentry process pose many difficulties for the
existing deterministic model, a significant portion of the research in the dissertation was
devoted to identifying limits of various estimation techniques that were considered for
th;s problem. [AReff. 1 021:p.
An adaptively determined, ad hoc, scalar finite memory or fading memory
parameter approach was selected. The motivation of this approach is based on the fact
that the existing reentry theory attempts to group all unmodeled parameters as a constant
with the ballistic coefficient (CDA/m) over some short trajectory span. [Ref. 102:p. 11]
In order to extend the existing orbit determination technique into the satellite
reentry problem, reference [102] defined a reentry dynamics mode! and developed the
weighted least-squares differential corrector structure currently used in the deterministic
model. The estimator update and propagation equations of the differential correctoi
structure for an infinite memory formulation are summarized as follows:
191
Prwpagations betwe epJchs:
State: integrate with initial conditions x.(t.), from the epoch at t.
A(t) = f(x(t),) (179)
to obtain xm(t. + 1) at epoch t,.Covariance:
Smi, 4.. d(t~,,,,t 1,,) S•,.,, •(tm.,itm)T (180)
Update at next ecpx&h:
State:I
X1,.q1(tm.. 1) = x.(t,.÷1) + •_ (xt~) 181)
i-1
Covariance:
S T R (n,-IT())- (182)S.el,nil = (sm+ ,' +TTR '
Recall that 1 is the number of iterations required to satisfy convergence and T(.) -R(,,) T(,) is evaluated from the reference trajectory on the final iteration, 1. [Ref.
102 :pp. 33-34]
Equation (179) is a nonlinear dynamics model, where f(x(t), t) is a deterministic function
of the state variables and is a continuous function of time. The variable, Ux(t), from
equation (181) is a "seeked" correction that will minimize a weighted quadratic cost
function of the observation residuals. Iterations of the differential corrector continues
until the observation residuals convergence criteria is satisfied. Equation (180) is the
initial state covariance matrix that remains constant until the iterative process has
converged. The quantity 4(tm+i,t.,) from equation (180), is the state transition matrix.
The final iteration in the process yields equations (181) and (182). R(.) and Tf() trom
equation (182) are respectively the matrices of the observation noise covariances and the
observation index being processed. [Ref. 102:pp. 20-29] As previously mentioned:
192
Application of this batch processing algorithm to reentry estimation has oftenresulted in poor estimator performance. This is largely due to the more significantnon-linear dynamics of reentry and the use of a deterministic and sirniplisticdynamics model ... in an uncertain dynamics region. [Ref. 102:p. 31]
In the development of the reentry dynamics model, the author chose an eight
dimensional state vector defined as [Ref. 102 :pp. 41-42]
1,1 =x
X2 tX
X3 Y
x(t) = (183)X5 z
16 =t
x7 = BPo
X8 Q
where
, = velocity components (184)
x,y,z, = position components (cartesian)
B = ballistic coefficient (CDA/m)
Po = sea level air density from equation (10)
Q = density s 'ale height (RT0 / M0g0') from equation (10)
M, = molecular weight of air
go = acceleration of gravity at Earth surface
To = atmospheric temperature at sea level
R = gas constant
193
H' - altitude (g/goj- from equation (W0)
g = lhral gravitational acceleration from the geopotential model x,y,zcoordinates
go reference geoid level gravitational acceleration
Ho = geocentric altitude (R-P)
R = local radius position relative to the Earth center (x? + y2 + z2)1n
R, = Ro (1 - f) [1 - (2f- P) cos " b] 1 re
f = flattening factor of the reference geoid whose elliptical shape is consistentwith J 2
65 = local latitude ( cos1 [ (x2 + y2)"2 / Ro )
Ro == radius of the reference geoid at the equator
The exponential atmospheric density model, equation (10).. was used in the
reentry estimator because of: [Ref. 102 :p. 37]
I. The reduced mathematical complexity.
2. The availability of continuous valued density and partial derivatives of thedensity along the reentry trajectory.
The eighth state variable, Q, was chosen since it is slowly changing at the
reentry altitudes ( < 100 kin). For this applical;on, the initial value, Q = 7.0031 kin
and its covariance were derived using a least squares fit to the base density values of the
altitude layers from the U.S. Standard Atmosphere 1962. A.!so, an Earth Centered
Inertial (ECI) coordinate frame was chosen to minimize the complexities of the
observation noise covariance matrix, R(,). [Ref. I02:p.40]
194
Both aerodynamic and geopotential terms within the deterministic function of
state variables, f(x(t) , t), are included in the estimator dynamics model. The dynamics
equation is given in the form [Ref. 102.pp. 42-44]
.ti ý --'x l = " 1 -2:
-t2 = fd + f.9 8j
-3=X4
-' = fd,+ fs (185)
X6 fd -
t7 0
.o 0
The aerodynamic acceleration x,y,z components from equation (185) are
derived from equation (6)
fd= -IBp VA (.t +•wy) (186)2m
1= -Bp VA(.-x) (187)
1 Bo VA (188)
The geopotential acceleration terms from equation (185) were derived from
the Smithsonian Astrophysical Observatory SAO-III Earth Model using the model
parameters and zonal and tesseral coefficients
195
CO aosc VVx - sin a/vv (189)
f, ' sin a, VV + cos c, (190)
fS, W /Z (191)_--
where
Of = Right Ascension (RA) of the Greenwich Meridian
The gradient terms from equations (189) through (191) are defined as
VI, - F, (C. + S" (192)010 ft-0
S -- =C,- - + S, (193)
a-•-+ a (194)
where
COM = zonal harmonic coefficient
San = tesseral harmonic coefficient
U: - GM &? P'"(56) cosGmX.) (195)
196
_" GM&4 P,"(8) sin(,nl) (196)
G = universal gravitational constant
M = mass of Earth
Ro = mean equatorial Earth radius
R = distance from Earth center
P.1 = associated Legendi.e polynomial of degree n and order m
= z/R (z -- coordinate)
X = longitude
As previously mentioned in Chapter II, in the region below 120 kim, the first-order zonal
harmonic, J7, may attain a magnitude approaching that of the atmospheric drag. Based
on this fact, in the development of the actual estimator and simulation, only the central
gravity, Coo, and Earth oblateness (J2), C20 , terms were used in the reentry altitude
regions.
Numerical simulations using simulated reentry data derived from a realistic
"truth model" were conducted on the basic estimator structure and dynamics model to
quantify its performance.
These analyses examined the effects of mismatch between the truth model and theestimator model dynamics, accuracy variations on the angular observations,multiple orbital observation locations and variations of the geometric relationshipsbetween the observing satellites and the reentry trajectory. [Ref. 102 :p. 14]
197
A series of Monte Carlo analyses were conducted which showed the model dynamics as
the most important factor impacting the estimator performance. This was accomplished
by:
1. Considering the velocity estimates and mean bias of the trajectory position vsthe magnitude of the standard deviations from the estimator computed statecovariance matrix.
2 Comparing the magnitudes of the standard deviations from the estimatorcomputed state covariance matrix to those derived in the Monte Carlo samples.
In the search to find possible solutions to the problems associated with the
estimator performance, reference [102] reviewed and discussed the limitations of several
model compensation methods that are relative to the reentry application, including: [Ref.
102:p. 15]
1. Adding a pseudo-noise compensation to the model dynamics
2. Adaptive estimation methods
3. State covariance deweighting techniques
The author developed a fading memory differential model compensation method based
on the fact:
... the previous numerical results indicated, the fundamental limitation of the infinitememory estimator formulation with deterministic dynamics is the biased estimatorsolutions which occur when processing true reentry dynamics. With i) exactdynamics ii) an upper limit on the time span of valid linearization, and iii) alower limit on observation accuracy (greater than or equal to 10'5 radians),acceptable estimator performance is available in terms of bias and RSS/(ONESIGMA) Ratio. [Ref. ,02:p. 112]
198
Since the variations in global atmospheric density and the dynamic changes
of the reentry body pose such a virtually intractable problem within a deterministic set,
improving the linearized estimator performance is based on the impact of uncertainties
within the ballistic coefficient and the atmospheric density [Ref. 102:pp. 112-113].
Improvement of the estimator was achieved by an adaptive determination of an ad hoc
scalar multiplier, -y. A finite memory on the processing of earlier observations is
implemented by multiplying the parameter, -y, to the terms of the state covariance matrix
prior to an observation update. At each update along the trajectory, the size of the
change in state variable, 6x, is examined. Each 6x, magnitude is then compared to the
magnitude of the standard deviation of their respective terms within a "deweighted"
covariance matrix which is computed by the estimation algorithm. As with the infinite
memor, estimator model, a series of Monte Carlo analyses were conducted to assess the
model's ability to estimate an anticipated true reentry dynamics trajectory. [Ref. 102 :pp.
15-161
Finally, reference [102] demonstrated the ability of the fading memory
method to provide a tangent plane projection of Earth impact locations as shown in
Figure 52 [Ref. 102:p. 192], by using bias magnitudes within the standard deviations of
the deweighted state covariance matrix.
The standard deviation of the position error from the deweighted state covariancematrix provides a good definition of the uncertainties in the estimated Earth impactlocations, thus it can be used to define a search area for recovery of satellitedebris. These results illustrate the viability of the method to estimate decayedsatellite impact locations and uncertainties significantly improved over existingastrodynamic applications. [Ref. 102:p. 16]
199
N X =TRUE IMPACT
a= 9.9km b= 2.78kmn one sigma
20 MEAN IMPACT
69.7 N. L AT61.5 E. LONG
10
TANGENT PLANE IMPACT ERROR ELLIPSES
Figure 52: Tangent Plane Projection of Earth Impact Location[Ref. 102]
200j
The primary conclusions and recommrLndations from the dissertation are:
[Ref. 102:pp. 198-202]
1. Dynamics uncert6i1-,ies of the general satellite decay trajectories significantlyaffect the estimator performance as shown by the Monte Carlo simulationanalyses.
2. Angular observation accuracies with standard deviations less than l0r radiansinduce significant error in the state estimate vs the standard deviations of thestate covariance matrix in deterministic dynamics models.
3. A recursive formulation of the estimator is recommended that uses a short timespan between the epoch or trajectory update point and tie observation(s) beingprocessed due to the anticipated dynamics uncer ?inties of reentry.
5. Multiple observations from more than one the orbital source are required toimprove the observability of the reentry and to provide higher data content oversimilar time spans in-order to achieve acceptable estimator performance in termsof bias and RSS/(ONE SIGMA) ratio.
6. An eight dimensional state vector provided superior estimation performance ascompared to the seven dimensiu.,ial vector used in the current determinisc-models. Performance improvement is achieved through simpler mathematics inthe dynamics model and continuous valued partial derivatives of the dynamicsover the trajectory space for a Taylor's series linearization.
7. The adaptive,ad hoc scalar fading memory parameter is easily incorporated intothe basic estimator structure.
8. A Monte Carlo derived impact covariance for the final propagation phase tcimpact preserves the integrity solution statistics over the non-observable finaiportion of the trajectory. Impact location uncertainties are on the order of oneto two magnitudes smaller than those available from current operationaltechniques.
9. Further investigations which vary the observational data rate and the timevarying character of true reentry dynamics are needed to examine the estimatorPerformance extensions Specifically, more accurate observations (much lessthan 10s radiani) with higher data rates and frequency variations as well asalternative measurements, such as range or range rate.
201
10. Further analysis of applying the fading memory method to very high altitudesand shallow reentry angles is recommended due to the fact that violent dynamicchanges under these conditions may result in divergent estimator performance.
11. Application of this estimation technique to a wide class of reentry trajectoriescould provide a large empirical data base to improve the estimator dynamicmodel. Dynamic model pseudo-noise compensation, statistical linearization, orhigher order filters investigations should be pursed.
2. Analysis of Tracking and Impact Prediction (TIP)
Reference [48] analyzed the accuracy of early TIP processing (Chapter II,
p.58) conducted by the USSPACECOM on 180 objects that decayed during the years
1987 through 1990. As part of the analysis, early TIPs (/ day to 3 hour predictions)
were compared to the final TIP. The time error for each TIP run was calculated and
compared to the ± 20% accuracy level claimed by the SSC at Cheyenne Mountain AFB.
Results from the comparison study indicate:
1. The decay prediction accuracy is usually, but not always within the claimedaccuracy level as shown in Figure 53 [Ref. 48:p. 37].
2. Che existence of a positive bias which indicates early TIPs are routinely laterelative to the final TIP.
"The author developed six mu'.tiple linear regression models that could be incorporated
into the TIP deca) procedures in an attempt to model out some of the positive bias found
in TIP decay prediction data. [Ref. 48:p. viii]
202
DECAY PREDICTION(Relative to 20 Percent Standard)
7.DAY 4.DAY 2 tOAY I-DAY 12-1W M i ~DECAY PREDICflON
Figure 53: Decay Predicwd Accuracy (By Year)[Ref. 48]
203
The data used in the investigation was collected from: [Ref. 4 8:pp. 15-17]
1. TIP Required Item Checklist--a manually kept chronological account of eachTIP processing.
2. Decay History--a computer generated log of each SP update that includes therun time, time of last observation, epoch time, epoch revolution number, B-term(ballistic coefficient), period and decay prediction.
3. Final Element Set--a listing of the final orbit parameters. The eccentricity andmean motion data from the listing were used in the investigation.
Only those objects that received the entire TIP update cycle (7 day through
final run) were used to accurately analyze the effects of each successive update and
prediction [Ref. 48 :p. 17]. In order to assess the accuracy level claimed by the SSC,
reference [481 used the data from the final decay prediction as the control by which to
compare the earlier predictions.
The final prediction was chosen as the control because it uses observations whichare closest to the actual impact point and is considered to be the most accurateprediction available. In order to further justify the use of the final prediction asthe control, a statistical anaJysis was also performed to directly compare the fewsighted reentry points, called Vis Obs, with the final prediction made by the SpaceSurveillance Center. [Ref. 48:p. 18]
Of the 180 objects studied, 93 were Vis Obs. Figure 54 shows the mean time
error of the final run time vs the Vis Obs [Ref. 4 8:p. 2 5]. 'The size of the final time error
standard deviation decreases as shown in Figure 55 [Ref. 48:p. 26). This decrease may
be correlated to the level of solar activity during the 1987-1990 time period. Solar
activity levels began to dramatically increase in 1987 and continued to increase through
the solar maximum (March 1990). The rate of change increase of the sunspot activity
204
MEAN TIME ERROR(Final Run vs VIS OBs)
-4-/
YEAR
Figure 54: Final Run vs Ws Obs Mean Time Error[Ref. 48]
205
mwmFINAL RUN TIME ERROR(Standard Deviation)
0
S196.19 97
YEAR
Figure 55: Final Time Error Standard Deviation[Ref. 48]
206
and solar flux during 1987-1988 was greater than in period from 1989-1990. According
to the author, the trend shown in Figure 55 may be related to the lesser rate of change
during 1989-1990. [Ref. 48:pp. 26-28]
The time error for each separate TIP run was calculated as the difference
between the predicted decay time for that run and the final run [Ref. 4 8:p. 18]. Figure
56 shows the graphic calculation results for the mean time error and time error standard
deviation [Ref. 48:p. 29].
In addition to the time error calculations, the location error was calculated
by taking the difference between the predicted location point for each run and the final
predicted point. The method used the mean motion (n) from the Final Element Set to
accurately determine the velocity for each TIP object. Since mean motion was
unavailable for the early TI1 runs, an approximation was made using the final mean
motion value for each TIP run. This introduced an error into the calculation of the
location error. However, because the location errors are very large (thousands of kin),
the eiror introduced by using the final mean motion instead of the actual mean motion
for that particular TIP run was considered insignificant. The mean motion value was
used to calculate the semi-major axis distance from the equation
a = 3 (197)
The velocity was calculated using the semi-major axis from the equation
207
MEAN TIME ERROR
3wI.I
U7- AY -DAY 3-CAT W +U * )i
TtRUN
TIME ERROR (1987-1990)(Sandard Deviation)
10
WIP RUN
Fligure 56: Mean Time Error And Standard Deviation (1987-1990)[Ref. 48]
208
v 2 42 ](198)
The object's velocity was then multiplied by the previously calculated time error in order
to determine the location error (difference between the predicted location point for each
run and the final predicted point). [Ref. 48:pp. 19-22] Figure 57 shows the graphic
calculation results for the mean location error and location error standard deviation [Ref.
48:pp. 32-33].
Multiple linear regression describes the relationship between several
independent variables and a dependent variable. The motivation for developing the
multiple linear regression model to eliminate the bias found in the mean time error was
based on the second objective of the investigation. This objective was to determine if
it was advantageous to initiate an OPREP-3 report (used to notify higher authority of a
potential reentry within 100 nm of the former Soviet Union border) earlier than the 6
hour mark. The first step in the process was to determine if the model in the form-
E(tA = Po + 1I3tI + 132t2 . P33t3 + 134t4 + 135t5 + PJ66 + P7tI (199)
where
E(tf) = expected value of the final decay prediction time
= early TIP decay predictions
= final decay prediction time
(3• = y-intercept and coefficients to be determined
209
Li
MEAN LOCATION ERROR(1987-1990)
0
7OO-
7 MAY 640AY '2DY "I-DY '3 U-M f+ 3-HR
LOCATION ERROR (1987-1990)
z
I-mp RUN
Figure 57: Mean Location Error And Standard Deviation (1987-1990)[Ref. 48]
210
could be found where the early TIP data could be used to approximate the final decay
prediction time. [Ref. 4 8 :pp. 22-23] The author states the following about equation
(199):
... multiple linear regression was then used to first determine if a model in the formshown.. .could be found to predict the final decay time. The results were an R-squared value of 1.0000 and a p-value of .0001. This means that at a significancelevel of .05 there exists a perfect linear relation between some of the independentvariable and the dependent variable where at least two of the / terms are not zero.The variance inflation values were all extremely large, indicating the independentvariables were all highly correlated and that a great deal of redurdancy exists inthe data. [Ref. 48:pp. 37-38]
Based on the above results, six separate linear models were developed in an effort to
approximate the final TIP with greater accuracy than the current process. Specific
characteristics used in the modeling include the following : [Ref.48:p. 38]
1. The first model uses only 7-day TIP data to calculate the expected value of thefinal decay preoiction time, E(tf).
2. One additional decay prediction data point is incorporated in each subsequentmodel.
3. All 180 TIP objects were used in the six models to calculate E(tf).
The six models are defined as
E(t = -0.116442 + 0.999064(t1 ) (200)
E:tA = -.0155478 4 0.49007(t,) + 0.951197(t2) (201)
E3(tA) = -.0082444 + 0.41692(tl) + 0.050001(t2) + 0.0908343(t3) (202)
211
E4(t = -0.053030 + 0.007471(,1) - 0.014395(t2) + 0.196518(t3) (203)+ 0.810492(4)
E(tý = -0.022322 - 0.004222(t1) - 0.005502(t2) + 0.046458(t3) (204)--0.049000(t4) + 1.012311(t5)
EetA = -0.008370 + 0.000553(tl) - 0.002944(t2) + 0.014541(' 3) (205)-0.009096(t 4) - 0.183413(t5) + 1.180378(t6)
where
t, = 7-day prediction time
t2 = 4-day prediction time
t3 = 2-day prediction time
t4 = 1-day prediction time
ts = 12-hour prediction time
t6 = 6-hour prediction time
Figure 58 shows the mean approximate error for the six regression models
[Ref. 4 8 :p. 40]. By comparing Figure 58 with the mean time error in Figure 56, the
regression models show a better mean approximation error than the TIP runs. The
conclusions and recommendations from the thesis are: [Ref. 48 :p. 4 2)
1. The decay predictions were much better in general than the reported ± 20 %.
2. The use of linear models in conjunction with the data generated by the TIPprocessing would allow the SSC to better predict final decay time by eliminationof positive bias in the data.
212
MEAN APPROXIMATION ERROR(1987-1990)
30/
40/
:19
40
3,2 ,3 4 5 0REGRESSION MODEX/,
Figure 58: Regression Model Mean Approximate Error (1987-1990)[Ref. 48]
213
3. The SSC should conduct a study of the current Special Perturbations model toattempt to better account for the level of solar activity and its effect on theatmosphere.
B. MONTE CARLO ANALYSIS OF SKYLAB'S IMPACT AREA
Reference [45] investigated the debris impact point dispersion area of Skylab. A
combination of Monte Carlo statistical analysis and parametric methods were used to
determine the three-sigma linits of the debris footprint. T'he investigation was conducted
by Martin Marietta Aerospace during the design arid development of the Teleoperator
Retrieval System (TRS) fbr the Skylab reboost/deboost mission. The dispersion analysis
was conducted to support the deboosting of Skylab to a safe oceanic impact area clear
of islands and routine shipping lanes. [Ref. 45:pp. 1-2]
The Monte Carlo statistical analysis is an efficient and realistic approach that can
be used to calculate the debris impact area because of the large number of input variables
and nonlinearities of the problem. Impact dispersion area boundaries depend on the
following factors: [Ref. 45:p. 2]
1. Entry dispersions
2. Relative flight path angle
3. Debris ballisdic coefficient
4. Breakup altitude
5. Environmental conditions (wind direction/magnitude and atmospheric density)
214
The entry dispersions were represented by a #, x 6 covariance matrix of velocity
and position errors at the entry altitude. For the deboost mission, the error sources were
the uncertainties associated with; [Ref. 4 5 :p. 2]
1. Accelerometers
2. Gyros
3. Computer errors
4. Initial alignment errors
5. Vehicle performance dispersions
The covariance matrix of vate variables was determined by conducting a trajectory error
analysis using a 1-sigma devialion of the errors and the nominal flight trajectory. In the
application of the Morve Ca~io analysis the major considerations were: [Ref. 45:p. 2]
1. The simulation of perturbed trajectories.
2. The modeling of thc satellite breakup.
3. The determination of impact points for each trajectory.
At the entry altitude, the Monte Carlo method required generation of random state
vectors from the 6 X 6 covariance matrix. Because the velocity and position errors are
correlated, the random error vector in an uncorrelated space was derived from a
transformation to a principle axes system. This vector is given by
215
e, = [nle1, Aes ... , q6ejT (206)
where
ni = a set of random numbers drawn from a normal distribution of mean 0 andvariance 1
ei = the square roots of eigenvalues of the covariance matrix
The original coordinate system random state vectors were derived from the equation
I?, (207)•- +I4,I1e,Il_-
where
4, = 6 x 6 eigenvector matrix that transforms the principle axes system into
the original coordinate frame
R_ = perturbed radius vector
VP = perturbed velocity vector
R. = norni al radius vector
V, = nona. .1 d velocity vector
In order to calcula c the dispersion area, the Monte Carlo analysis required a large
number of simulate( rajectories and random perturbed state vectors at the entry point
for each perturbed i. ial condition. [Ref. 45:pp. 2-3]
Additionally, th. Monte Carlo scheme required a fast and efficient computational
technique to calculat flight trajectories from several initial conditions. AnalyticaJ.
solutions, such as Loh's second-order theory (presented in Chapter lII) were not su .e
for footprint dispersion analysis because these solutions are valid for only certain portions
216
of the trajectory and entry conditions. The trajectory simulation used the equations of
motion of a nonlifting vehic!e in the form
211P
- V(V + WX)g (209)€' , .. zB+• (209)Y ny 2BP
O PV(V,)g1 (210)
where
V •- vf, +oy) 2 + (V, - cx)2 + (VZ) 2 (relative velocity) (211)
•,g ,gg = gravitational acceleration components
"11 V1, V, =nertial velocity vector components
x,y,z = radius vector components
W - Earth's rotation rate
BP = ballistic parameter or coefficient (W/CDA)
Equations (208) through (210) were integrated several times from entry to the impact
point for the Monte Carlo analysis. The integration step size used was as large as
possible due to a stability consideration. The system approached a dynamic equilibrium
condition which resulted in a numerical instability. This was a result of the opposing
gravitational acceleration and aerodynamic deceleration during the vertical portion of the
flight. A numerical search was conducted to determine the step size as a function of
217
satellite debris ballistic coefficient. [Ref. 45:pp. 3-4] Figure 59 shows th~e stable and
unstable regions used for the numerical integration [Ref. 45:p. 4].
Since the Monte Carlo analysis required the simulation of a large number of
trajectorics, an accurate computationally fast atmospheric density model was needed.
The authors chose the 1962 U.S. Standard Atmosphere that provided density values from
0 to 400,000 ft. [Ref. 45:p. 5]
Althcugh both static and dynamic global atmospheric models accounting fordiurnal, seasonal, and latitudinal variations are available, it is found that suchmodels are not suitable for the Monte Carlo dispersion analysis of satellitefootprints. A sensitivity study of density variation indicates only a second ordereffect on footprint dispersion. Furthermore, the results of Purcell and Barberyshow the downrange impact errors resulting from atmospheric variations are slight.[Ref. 45:p. 5]
The simulation assumed one breakup altitude where all the pieces had the same
initial position and velocity at breakup. By using a bounded parametric approach, the
smallest and largest debris ballistic coefficients were used to determine the largest
uprange and downrange footprint dispersions from the nominal impact point. Since the
ballistic coefficient will vary as it passes through the atmosphere due to variations of CD
and Mach number, the program used a generalized CD vs Mach number curve for
tumbling pieces to update the ballistic coefficient at various altitudes. This curve is
based on the results of a range safety study of Titan launch vehicle debris. [Ref. 45:p.
5]
For the footprint dispersion study, it is found that the transonic flow region, whereCD variation with Mach number is significant, occurs during the vertical descentof the debris with minimum effect on footprint dispersion. However, the variationof ballistic parameters with altitude affects the time of arrival of the debris on theground. [Ref. 45:p. 5]
218
140 -N
"-120-
"1I00 - -
"- 80- Stable Unstable
4-3ai 60E -
40-
cA__--
"-" 20ca
00 5 10 15 20 25
Integration Step Size (sec)
Figure 59: Numerical Stability Region[Ref. 45]
219
The latitude and loagitude (lat/long) impact points of the smallest and largest
Skylab debris were provided by the trajectory simulation for the various initial
conditions at the entry altitude. The Lat/Long impact points were statistically treated
which determined a three-sigma impact area boundary. The point estimate and
confidence level estimate of the impact points for any given confidence level, E, and
probability, ci, are the fundamental statistical quantities of interest.
These statistical computations relate the calculated longitude and latitude boundsfrom a finite sample to their true values corresponding to an infinite samplerequired for the Monte Carlo technique. [Ref. 45:p. 6]
By using a normal distribution approximation given by
N- a K,_ (212)(1 - a)
where K, is defined from the equation
-2 = (1 - ai at e223)
equation (212) provides the estimated number of samples required to estimate a good
point. A probabilistic statement given by
P.(O < 0C) = a (214)
P,(4d46 ) -(215)
where
0 = impact point latitude
0 = impact point longitude
220
was obtained from the frequency histogram and cumulative probability distribution of
Lat/Long impact points generated from the Monte Carlo analysis. For any specified
value of a, the object of the statistical analysis was the estimation of 0,. and 0.. [Ref.
45:p. 6]
1. Simulation Results
During the simulation to determine the debriL footprint dispersion, Skylab was
deboosted from a 170 nm circular orbit. The nominal entry and corresponding impact
point Lat/Long were respectively (30'S,30°W) and (49.86°S,34.92'E). The nominal
entry point state vector and the lower half of a symmetrical ECI frame covariance matrix
are shown in Table XVI [Ref. 45:p. 7]. By using a Gaussian distribution for the Monte
Carlo analysis, the initial state vectors were randomly selected from the covariance
nainrix. Specific-aily, 500 r1adoU eniry siaites were creaied by using 30(X) iauduiini
numbers drawn from the distribution.
To ensure that these numbers truly represent a normal distribution, their mean andvariances were determined and adjusted to be 0 and 1 within stringent tolerances.The number of samples were found to be sufficient because further increase insample size did not significantly effect the output variable distribution andprobability limits. [Ref. 45:p. 8]
Figure 60 shows a scattergram of the state vectors which define the entry flight path
angle [Ref. 45:p. 7]. The trajectories were then simulated from the entry points down
to tne breakup altitude for each randomly selected state vector. Figures 61 and 62 show
the results of a nominal run used to determine the flight characteristics of Skylab [Ref.
45:pp. 8-9]. From Figure 62, the following parameters vs altitude were presented:
221
Table XVI: ENTRY STATE VECTOR / ERROR COVARIANCE MATRIX[Ref. 45]
Nominal Entry Point
x x 17,648,032 ft Vx = -3238,58 ft/sec
y = -5,174,810 ft Vy 21011.96 ft/sec
z z -10,760,572 ft V z -14693,59 ft/secZ
ECI Error Covariance Matrix
x y z l V .V
x 5.76546E8
y -3.95374E9 2.72016EI0
z 2.79469'9 -1.92075EI0 1.35675EI0
V 4.86448E6 -3.34635[7 2.36298E7 4.11679E4
V -1.35955E6 9.34064E6 -6.59878E6 -1.14913E4 3.20968E3y
Vz -3.03260E6 2.08426E7 -1.47220E7 -2.56422E4 7.16010E3 1.59759E4
222
-0,820
-0.824-
-0%823 a
w. -0.832 6 "
e * * *
l S..-.
C 0 S* * 5 * •
0L- -0.868
.: -0.852 -:c
-0,8561 I _I I I I I .0 50 100 150 200 250 300 350 400 450 500
L Number of TrialsFigure 60: Relative Entry Flight Path Angle Scattergrani-'-'''••••••[Ref. 45
223
c3~ ess a a ga a eaIei *-,
400000
'Z 3000004-
• 200000 -
< 100000
0 ________... .______
0 50 100 150 200 250 30.90 Dyanmic Pressure (lb/ft 2 )
III I I I I I
0 5 10 15 20 25 30 350 MACH No.IIII I I
0 2 4 6 8 10
A Load Factor [(dv/dt)/g]I I I I I I !0 5000 10000 15000 20000 25000 30000
-I- Velocity (ft/sec)S I, I I IL L _I_ I JI
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
X Relative Flight Path Angle (deg)
Figure 61: Skylab Nominal Flight Characteristics[Ref. 45]
224
400 000 -BP :35 lb/ft2
Breakup Altitude
~! 3__ --04n-
Ir
K300 000
BPm
1 00 000ý
01
0 400 800 1200 1600 2000 2400 2800 3200 3600
Downrange (n ml)
Figure 62: Downrange Impact Point Variations[Ref. 45]
225
1. Dynamic pressuie
2.' Mach number
3. Load factor
4. Velocity
5. Relative flight path angle
In Figure 62, the 275,000 ft (84.8 kin) breakup altitude of the simulation is shown along
with the downrange variations for several debris sizes (denoted by ballistic parameter or
coefficient). The lat/long histogram and cumulative probabiiity distribution diagrams
respectively shown in Figures 63 and 64 [Ref. 45:pp. 11-12] indicate the two distinct
regions corresponding to the largest and smallest ballistic coefficient debris.
Additionally, these diagrams determine the, lat/long hnnndq where- ill debris is likely to
fall. [Ref. 45:pp. 7-12]
The downrange and crossrange debris impact dispersion distances were
calculated using the lat/long and azimuth angles at the nominal impact point by the
matrix equation [Ref. 45:p. 12]
226
"I . -I -4-. t A-
-4..9973?1.01 4.74 ** 4b*** *9*@**#**Se4-4..9R173F*OI 14 +*-4..94610-01 + %NALI.FJt 'ALLISTIC PARaAe4EIPR-4.ri%U4fv4)1 2 FmlPMas 10 u LIKdFTO*d-4.9149SF*fl1 1 a-4.9193SE#; 4 .-4.9037AP.()1 7S *4.
-4.AAAN14E4C) 1A5-4, 01 757 1 112 +4*4**
-4, 8 5697 ?0+ I S4
-4eA413AF*01 ?9-4.01257AIF'O1 13 LAQ(,FST KALLiSTIC PARAMETER-4.A1(J1cQE+1 9 n4Fl.R1's mP a 115 LK/FTS*2-4.79459E+01 4. +
-4.77900E4OI a-4..76341E*01-4.747ALF*OL 3 .
-4. 73Z222E01 2 +-4.7146!2E+01 0 +-4. 70103E*01 7 2
CLASS COMUILAT IVE C13M1ILAT IYFBrAJNAiR I ES FREQ11ENCY fýAE0IFNCY PLOYl
0 0.5 1.0-5001242E+01 ------409973?E4O 1 .4.7900 4*-4,9A1 73F.01 .4.920 eZOse***4*
-4.9&6&L3E4Ol .4,Q700 **404S*4**-4..95054,E+01 .4.9900 *440s484s4*
-4091495E40l 0 50000 40.s4ee*a*
-* 4 9193SE*O1 19S0900 ... s.**e~~e
-4. 90376E*01 SFI400 *****44404@*
-4. ASRI 6pF01 * T4.90 ,*.g**,.s.*e..Os-4.17Z57F*01 *@A 100 *04444S0*S4*S*SS*4-4.AN5A97E4%01 *93500 .4*,44S.4**,**S**4b
-4.84'138E+01 .94$400 *S*34440*4***46*4S*
-4. f2576F+01 '97700 ~**4Q*49**4*4*S*$SS*-4.81019EF+01 .9bO V44*****4**4k**%4Sb0*44
-4. 79459E*O1 .99000 *0@44*44*44***0S4SSO
-W.77U0e*01 qe100 *444.b**4**44**4444444g
4. 7634.1E*01 q99300 *4e.*ees.*444.4'a*Ss*-4 .747A 1 p.40 .996.0 *44* 4***4***0444S44
-4. 73222F+Ok .99800 ,e**4~4***es9@**s44s
-4. 71.6?E.01 .99400 .4*4S*.*4.*.~4*4,*4.
-4.70103E*01 1.00000 **44**48440S4444*444
Figure 63: Latitude Histogram /Cumulative Probability Distribution[Ref. 451
227
LrIN11,Ttinr STATISýTI(.S
CL A H140-lT.14AM (IF
PRfhI 1% G I F % F R F 1IF00. Y F Ro IFN~C.Y
2.9(JN3qF*Ol 1RI3.o3R7QE*fll 11%23.16919F<-01 66 we~**UUS
3.29959F+0)1 16%'* SMALLFST tbAL1ISMI PARAMqETERt
3.42994E+01 13 lFtiRIS lIP z10 LlWFISS?
3.56O3RE+01 1 4
3.69077E-01OI03.AZII.1E*Ol 1
3.9515AE*0C)'#
4.21235F*O01 44.34275F+01~ M
4.4711.4F*O1 1h7*S8 s.,$ea**O*Sw***
4C.60354F+.OIo
4.73393E*OI 2o *~~ LAA.'FST bALLISTIL PARANIFTER
4.RA433E+01 Q OFORIS1 UP z MI' LB/FTOU2
14,99472E*01 ?'
5.1?'iL2F*01A
CLass 0110101.AT IVF CUMULATIVE9LOW0AR ES FitE04IE84L FREQtUENCy PLflT
(0 0.5 .
2.64161F.O1 * ----- ----
Z*0T@OOE*Ol .arOS700
1.90639EOI .t?4200 e~**e
3.03e?9F.O 1 .194.003.1l6919F+01 .41,1001) ,*ee9**SW
3.29959E*01 .47&00 4 **8**SSO**
3.4299RF01 .414900) **9****~
3.56038F+01 *g.44 *.*e4**C*oo*
3.69077E*OL .4.96.100 4*C***
3.821171F+01 .4.QROO .. *******e
3.9SIS6F+01 . ,)Oflu0 ... e***O*
4 on 196F +()I ~%Oom)(4.2123'SE401 .5'4.?)) *S***oS*....
14.34275F.OI .7?Aon S***9**O***~~
4.4?13 4E*01 Aqso()(
4.73393F*Cp1 ,9R 101 ** .e*.*e e
4.A6433E*0O .490Mf~f)
S*?S5?'+0 IE( .*94)
Figure 64: Longitude Histogram /Cumulative Probability Distribution[Ref. 45]
228
V, E -C1S0 0,,SSS0,V 5XCO, - CXS 4O5ON CXC4N XP - XN
T, C)I*S~t-SXLSONCON -_SXS4),SON-C1C8 SXC46 P YN 26
A C CssNCN C- ,,,-S c s4 ZP,- ZN
where
= orthogonal coordinate system established at the nominal point
= axis tangent to the ground trace at the nominal point
= axis along the radius vector
= axis completes a right-handed triad
nominal impact point latitude
ON =nominal impact point longitude
X = azimuth angle ( sin'[cos i / cos ON] )
XN = RNCONCON
YN = RNCONSON
ZN RNSON
x= RpCtpC~p
Z" RpSpp
S =sin
C =cos
229
N nominal
P = perturbed
Figure 65 shows the downrange and crossrange dispersion distances from the nominal
impact point calculated from equation (216) [Ref. 45:p. 14]. As shown in the figure, the
crossrange dispersions were relatively small as compared to the downrange dispersions.
In order to determine the effect of the breakup altitude on the dispersion area,
Monte Carlo simulations were conducted at various altitudes from 200,000 to 350,000
ft (61.7-107.9 kin). in this altitude range, the reentry object experiences large thermal
and structural loads. Figure 66 shows three-sigma downrange and uprange dispersion
for the largest and smallest debris [Ref. 45:p. 151.
This figure is useful for estimating the total footprint disrersion resulting frombreakup of smaller pieces at higher altitudes and heavier pieces at lower altitudes,thereby accounting for the incertainty in breakup altitude. [Ref. 45:p. 1141
In their conclusions, the authors note significant footprint dispersion
variations for breakup in the 200,000 to 350,000 ft altitude region. Furthermore, they
state that the comprehensive Monte Carlo analysis is very useful and appropriate in
determining impact dispersion areas of a spacecraft or discarded portions of a launch
vehic' [Ref. 4 5:p. 15]
230
8.0-
6.0-
E*
4.0-
Nominal Impact Point
2, - '-
a .0.
v 4,0-L
o '* *
S= $I .. *
o $ $**$$****$$ *
S-2.0
-4,0 Ij 1I Ijj-
-400 -200 0 200 400 600 800Dipersin (mi Downrange Dispersion (n! mi)Uprange
Dispersion (n mi)DonagDipron( )
Figure 65: Down Range And Crossrange Dispersions
[Ref. 45]
231
BP 35 lb/ft 2
-400
Breakup
BP 1 10 lb/ft2 Altitude (1000 ft) BP 135 lb/ft2
350
300
-3 Sigma Limit 250+3 Sigama Limit
200
J ~~150 1 1 1 1 11-800 -400 0 400 800 1200 1600 2000
Uprange Dispersion (n mi) Downrange Dispersion (n mi)
Figure 66: Downrange And Uprangc Dispersions Three-Sigma Limits[Ref. 45]
232
VI. CONCLUSIONS AND RECOMMENDATIONS
A. CONCLUSIONS
The primary goal of this thesis was to identify the "state-of-the-art" of orbit-decay-
induced uncontrolled reentry and impact prediction methods, with an emphasis on the
physics of the final few revolutions to impact. This was accomplished through a
comprehensive literature survey from the 1950's to the present of unclassified military
and civil databases. The survey indicated that there is some significant foreign work
being done and much of it was not available to the authors, in English translation, and
thus was not included in this survey. Also, the literature survey reflects the changing
scientific terminology over the course of several decades and it is especially noticeable
in the different forms that the common variables take in the numerous equations
presented. The authors did not make any attempt to use the standard AIAA
astrodynamics nomenclature or to standardize the equations in any other way.
The principal conclusion of this thesis is that the current uncontrolled reentry and
impact prediction methodology, used in the U.S. and abroad, is based on analysis which
is 30 or more years old. This conclusion is based on the fact that the U.S. method takes
its roots in the works of Brouwer (1959) and Allen and Eggers (1957), and that the U.S.
method is the accepted international standard, as shown by the literature survey.
While conducting the literature search dating back to the 1950's, the authors
noticed a definite trend, through the years, in the focus of published material related to
233
reentry. During the timeframe from the late 1950's until the mid 1960's, the emphasis
in the literature was primarily devoted to understanding and describing the physics of
reentry, specifically as it pertains to controlled ballistic reentry of Mercury and missile
type vehicles. With the exception of Sputnik IV, very little of the literature surveyed
during this timeframe was strictly devoted to uncontrolled reentry. However, during the
derivations of their analytical reentry theories, some early pioneers such as Chapman and
Loh investigated the shallow orbit-decay-induced recntry. With the physics of reentry
fairly well understood, starting in the late 1960's and continuing into the 1980's, the
emphasis in the literature had shifted to controlled, gliding reentry in order to support
the launch of the Space Shuttle. In 1965, Eggers and Cohen stated:
Significant advances have been accomplished in the science and technologyappropriate to atmosphere entry of spacecraft during the eight years since thelaunching of Sputnik I. This progress is illustrated by the successful entry fromEarth orbit ot thle manned Vostok, Mercury, Vosknod, and Gemini spacecraft., forwhich the problems in orbital entry, such as high convective heating rate and loadand communication blackout, were successfully over come.. Although the firstApollo vehicle entry has yet to be demonstrated... Much engineering work for thisvehicle remains to be done; however, the fundamental research activity associatedwith Apollo is being reduced in favor of that associated with missions and vehiclesof the more distant future. [Ref. 75:pp. 339-340]
Uncontrolied reentry briefly came back into focus in the late 1970's and early 1980's
with the reentry of Cosmos-954 and Skylab and continued throughout the 1980's in
varying degrees where the reentries of Cosmos-1402, 1601 and Salyut-7/Cosnios-l.686
continued to spark a flurry of literature from the European Space Agency.
The literature survey of recent publications shows a strong reliance on work done
in the 1950's and 1960's as a basis for "extensions" or "modifications" of pre-existing
234
methods used for prediction of reentry and impact. Although reliance on pre-existing
methods is not in and of itself flawed, the modem works often subscribe to the same
assumptions which the original authors were forced to make because of hardware or data
limitations of their time. This serves to ptlrpetuate the inherent limitations of the
method's ranges of applicability and validity. For example, it has been shown [Refs.
14,16-17] that some very important work done in the 1950's, such as Allen and Eggers
[Ref. 65], which is still routinely referenced in recent publications, is inherently flawed
due to assumptions made under conditions of little or no data. This particular reference
is especially important since it is the basis for most of the modem reentry heating work.
It becomes even more significant since the literature survey has shown the coupled
dependence of reentry breakup to reentry heating and dynamic load effects. Refcrence
[421 describes the limitations imnOSeA_ by these assumptions through mathematicl proofs.
This is further substantiated by reentry breakup observations [Ref. 86] which indicate that
the classical convective heat transfer equations, when applied to breakup anialysis,
consistently underestimate reentry survivability.
Of the various "extensions" to the current reentry theory, of which the NORAD
method is recognized as the international standard, there does not appear to be any one
method which is singularly superior to the others. However, it is the opinion of the
authors that the ESA FOCUS program merits special attention for further research. This
conclusion is based on the fact that this prograi, contains a very sophisticated method for
dealing with the drag coefficiei t, C,,, and could easily incorporate other higher-order
effects occurring in reentry, such as attitude generated lift, rotation of the orbital plane
235
(due to atmosphere rotation at attitudes < 60 km), linear variations in the ballistic
coefficient in the final few days of orbit and others as mentioned previously in this thesis.
Numerous works cited in the literature survey also describe computing power and
computer time constraints as critical parameters of the problem. These hardware
limitations then forced one or more simplifying assumptions to be made in order to deal
with the technology limitations of the day. Subsequently, much of the literature survey
shows work which has limited applicability for the problem of concern.
Additionally, there is a general lack of consensus in the literature as it pertains to
orbit-decay-induced uncontrolled reentry. The very definition of "reentry" remains vague
and often is defined differently by investigators attempting to do similar research. Many
investigators have examined various aspects of the problem; however, when surveying
the literature, a common approach and standardized starting point for solving this very
dynamic problem is not apparent. A good example to illustrate this point is the effect.
of angle of attack, a, on the ability to accurately predict uncontrolled reentry and impact.
The ability to properly model the uncertainties in the reentry body configuration such as
changes in area, mass, and attitude is a very difficult problem. As a result of these
changes, three-dimensional angle of attack becomes an exceedingly difficult parameter
to characterize especially when the reentry body is undergoing rapid configuration
changes due to ablation and structural deformation. However, in order to properly model
the aerodynamic coefficients for lift and drag, angle of attack must be considered.
Specifically, lift as a function of angle of attack not only affects the trajectory (location
of impact), but it also affects the altitude of breakup (survivability a~id dispersion area)
236
since it reduces the thermal and structural loads on the vehicle. However, in most cases
where angle of attack is investigated, this coupled effect is not mentioned nor pursued.
Even mure likely in the literature, the reentry body is modeled as a point mass where
angle of attack is not considered or lift is assumed to be zero or negligible.
Another observation from the survey is that no comprehensive sensitivity and error
analysis has been conducted in order to determine quantitatively the effects of critical
parameters/variables on impact prediction. For example, numerous sensitivity and error
analysis studies have been conducted on atmospheric density models uncoupled from
other critical parameters. However, a coupled analysis of all of the critical parameters
could determine which variables contribute most significantly to the overall accuracy of
the various impact prediction methods.
In addition to the author's observations and conclusions described in the previous
paragraphs, a summary of ,pecific conclusions from the literature survey are:
1. T1here is no clear definition of when reentry occurs. Typically 120 km or anorbital period of 87.5 minutes or less is used in the various models to define thestart of reentry. However, observations and post flight analysis indicate a rangeof altitudes where "reentry" actually occurs.
2. There is a lack of observation data in the reentry regime due to a lack of globalsensor coverage. This lack of data significantly adds to the uncertainty of theproblem.
3. The current deterministic dynamics model appears to be inadequate forprocessing the true physics of reentry.
4. The lack of observational data coupled with the inadequate knowledge of thetvue physics of uncontrolled reentry significantly increases the uncertainty of theproblem.
237
5. Reentry is most likely to occur within several tenths of a degree of the minimum.altitude (height above the ground) point in the orbit, which is not necessarily thepoint of perigee. [Ref. 99]
6. T.e uncontrolled reentry is three times more likely to occur in a latitude bandequaling the inclination than in an equatorial band (0 to 10 degrees), with theexception of polar orbits where an equatorial reentry is most likely to occur.[Ref. 99]
7. The impact location (downrange and crosstrack) is affected by: [Refs. 45,101]
(a) the rotation of the atmcsphere, starting at an altitude of 30-60 km
(b) the debris ballistic coefficient
(c) the entry dispersions (velocity and position errors at the entry altitude)
(d) the relative flight path angle
(e) the breakup altitude
(f) the environmental conditions such as wind direction/magnitude and- . . .I .- 1- J • "ati-nospl-tel-ic ueflsltY
8. There appears to be linear variations in the ballistic coefficient in the final fewdays prior to reentry. [Ref. 92]
B. RECOMMENDATIONS
Based on the above conclusions, the following recommendations are proposed:
1. As mentioned in Chapter I, there was no attempt made to standardize thenomenclature and variables of the equations presented in this thesis, inaccordance with AIAA astrodynamics standards or any others. Based on theliterature survey, there clearly exists a need for such standards, especially whendealing with work spanning the course of several decades.
238
2. This literature survey was conducted through unclassified sources, written inEnglish, only. A survey of classified and foreign language sources should beconducted.
3. The primary focus of this thesis was a thorough survey of the physics ofuncontrolled reentry. Throughout the survey, the authors noted severaloperational issues which should be addressed in follow-on research. These issuesinclude human factors (orbit analyst experience), sensor bias and coverage, andimproved international information exchanges.
4. Several reentry/impactprediction methods were presented (LIFETIME, FOCUS,and NORAD) in this thesis. A side by side comparison of these methods usinghistorical data should be conducted in order to compare and contrast theiroverall accuracies and efficiencies.
5. A sensitivity and error analysis ot critical parameters including atmosphericdensity, aerodynamic coefficients, initial conditions (vehicle area, mass, attitude,position and velocity), thermal and dynamic loads, and breakup phenomenashould be conducted in order to improve the current dynamics model and tofocus future research efforts.
6. In view of the lack of global sensor coverage, especially in the southernhemisphere, high priority reentry events may be better predicted through a Joi"tcooperative effort using Navy and Air Force assets as mobile supplementalobservation and tracking stations.
7. Since uncontrolled reentry is characterized by numerous rapidly changingcritical parameters, which are ill-defined, stochastic and statistical methodsshould be applied to current reentry models to better analyze the sensitivity ofthe various uncertainties associated with this problem. This could serve as an"operational tool" to help improve the prediction accuracies of the currentmodels while the physics of tl e reentry process is being more thoroughlyinvestigated.
8. The authors strongly recommend future cooperative research between the NavalPostgraduate School and the AFSPACECOM/USSPACECOM in order to solvethis lawgely unknown and highly dynamic process.
239
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248
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